2.1 Directed graphs

In this section, we fix notation regarding the basic concepts of graph theory.

A directed graph is a pair $G = (\mathcal V_G,\mathcal A_G)$ , where $\mathcal V_G$ is a set and $\mathcal A_G$ is a subset of $\mathcal V_G\times \mathcal V_G$ such that

$$\begin{align*}(v,v')\in \mathcal A_G \ \Rightarrow\ (v',v)\notin \mathcal A_G.\end{align*}$$

We will typically simplify notation and write $\mathcal V$ and $\mathcal A$ instead of $\mathcal V_G$ and $\mathcal A_G$ . An element of $\mathcal V$ is called a vertex and an element $(v,v')$ of $\mathcal A$ is called an arrow from v to $v'$ . We shall also say $v'$ is the head of the arrow $(v,v')$ while v is its tail. Given $a\in \mathcal A$ , we write $t_a$ for its tail end $h_a$ for its head. As usual, the picture

will mean that $(v,v')\in \mathcal A$ . A loop in G is an element $a\in \mathcal A$ such that $t_a=h_a$ . We will only consider graphs with no loops, so, henceforth, this is implicitly assumed. We also assume G is finite, i.e., $\mathcal V$ is a finite set.

Given a subset $\mathcal V'$ of $\mathcal V$ , the subgraph $G'=G_{\mathcal V'}$ of G associated with $\mathcal V'$ is the pair $(\mathcal V',\mathcal A')$ with

$$\begin{align*}\mathcal A'=\{a\in \mathcal A:t_a,h_a\in \mathcal V'\}.\end{align*}$$

In terms of pictures, $G_{\mathcal V'}$ is obtained from G by deleting the elements of $\mathcal V\setminus \mathcal V'$ as well as all the arrows starting at or heading to an element of $\mathcal V\setminus \mathcal V'$ . It will often be convenient to write $G\setminus \mathcal V'$ instead of $G_{\mathcal V'}$ .

Let $\mathscr P(\mathcal V)$ be the power set of $\mathcal V$ and $\pi :\mathcal A\to \mathscr P(\mathcal V)$ be given by $\pi (a)=\{t_a,h_a\}$ . The (nondirected) graph associated with G is the pair $(\mathcal V,\mathcal E)$ , where $\mathcal E=\pi (\mathcal A)$ . The elements of $\mathcal E$ will be referred to as edges. By a (nondirected) path of length $m\in \mathbb Z_{\ge 0}$ in G, we mean a sequence $\rho =e_1,\dots ,e_m$ of edges in $\mathcal E$ such that

$$ \begin{align*} \#(e_j\cap e_{j+1})=1 \ \ \text{for all}\ \ 1\le j<m \ \ \text{and}\ \ e_{j-1}\cap e_j\cap e_{j+1}=\emptyset \ \ \text{for all}\ \ 1< j<m. \end{align*} $$

This is equivalent to saying that there exists an underlying sequence of vertices $v_1,\dots ,v_{m+1}$ such that $e_j = \{v_j,v_{j+1}\}$ for all $1\le j\le m$ . This sequence is unique if $m>1$ . If $v_1=v_{m+1}$ , we say $\rho $ is a cycle based on $v_1$ . In that case, if $m=\min \{j>1: v_j=v_1\}$ , we say $\rho $ is an m-cycle. Note there does not exist m-cycles for $m\le 2$ .

We shall often write $\rho =e_1\ldots e_m$ instead of $\rho =e_1,\dots ,e_m$ and set $\ell (\rho )=m$ . We also write $e\in \rho $ to mean that $e=e_j$ for some $1\le j\le m$ . Suppose $\rho '=e_1'\dots e_{m^{\prime }}^{\prime }$ is another path such that $e_m\cap e^{\prime }_1\ne \emptyset $ and either

$$ \begin{align*} e_m = e^{\prime}_1 \quad\text{or}\quad e_{m-1}\cap e_m\cap e^{\prime}_1 = \emptyset = e_m\cap e^{\prime}_1\cap e^{\prime}_2. \end{align*} $$

Then, the sequence obtained from $e_1\ldots e_me^{\prime }_1\ldots e^{\prime }_{m'}$ after successive deletion of any appearance of a substring of the form $ee, e\in \mathcal E$ , is a path which we denote by $\rho *\rho '$ . The path $\rho ^- := e_m\ldots e_1$ will be referred to as the reverse path of $\rho $ . In particular, $\rho *\rho ^-$ is the empty sequence.

If $m=\ell (\rho )>1$ ,

$$ \begin{align*} v\in e_1\setminus e_2, \qquad\text{and}\qquad v'\in e_m\setminus e_{m-1}, \end{align*} $$

we say $\rho $ is a is a path from v to $v'$ . If $\ell (\rho )=1$ , say, $\rho =e_1=\pi (a)$ for some $a\in \mathcal A$ , $\rho $ can be regarded as a path from $t_a$ to $h_a$ and vice versa. We let $\mathscr P_{v,v'}$ be the set of all paths from v to $v'$ and $\mathscr P_G$ be the set of all paths in G. If $\rho \in \mathscr P_{v,v'}$ and $\rho '\in \mathscr P_{v',v"}$ , then $\rho *\rho '\in \mathscr P_{v,v"}$ .

A subpath $\rho '$ of $\rho $ is subsequence such that

$$ \begin{align*} e_i,e_j\in\rho' \quad\text{with}\qquad i<j \qquad\Rightarrow\qquad e_k\in\rho' \quad\text{for all}\quad i\le k\le j. \end{align*} $$

We say $\rho $ is a simple path if no subpath is a cycle. If $\rho =e_1\ldots e_m$ is a path from v to $v', e_j=\pi (a_j)$ , and $m>1$ , the signature of $\rho $ is the element $\sigma _\rho =(s_1,\dots ,s_m)\in \mathbb Z^m$ given by

$$ \begin{align*} s_1 = \begin{cases} -1,& \text{if } v=t_{a_1},\\ 1,& \text{if } v=h_{a_1},\end{cases} \qquad\text{and}\qquad s_{j+1} = \begin{cases} s_j,& \text{if } t_{a_{j+1}}=h_{a_j} \text{ or } t_{a_j}=h_{a_{j+1}},\\ -s_j,& \text{otherwise,}\end{cases} \end{align*} $$

for all $1\le j<m$ . If $m=1$ , the signature will be $1$ or $-1$ depending on whether it is regarded as a path from $h_{a_1}$ to $t_{a_1}$ or the other way round, respectively. We shall say $\rho $ is monotonic or directed if $s_i=s_j$ for all $1\le i,j\le m$ . In that case, if $s_j=1$ for all $1\le j\le m$ , we say it is increasing. Otherwise, it is decreasing. If $\rho $ is increasing, we set $h_\rho = h_{a_1}$ and $t_{\rho }=t_{a_m}$ . If it is decreasing, then $t_\rho = t_{a_1}$ and $h_{\rho }=h_{a_m}$ . If $s_{j+1}=-s_j$ for all $1\le j<m$ , we say $\rho $ is alternating. Clearly, $\sigma _{\rho ^-}=(-s_m,\dots ,-s_1)$ . We shall refer to a monotonic cycle as an oriented cycle. We will denote by $\mathscr {P}^+_{v,v'}$ (resp. $\mathscr {P}^-_{v,v'}$ ) be the set of increasing (resp. decreasing) monotonic paths from v to $v'$ . For instance,

On the other hand,

but is neither in $\mathscr {P}^+_{v_1,v_3}$ nor in $\mathscr {P}^-_{v_1,v_3}$ .

A graph G is said to be connected if, for every pair of vertices $v\ne v'$ , there exists a path from v to $v'$ . If G is connected, we can consider the distance function $d:\mathcal V\to \mathbb Z$ defined by, $d(v,v)=0$ for all $v\in \mathcal V$ and

$$ \begin{align*} d(v,v') = \min\{\ell(\rho): \rho\text{ is a path from } v \text{ to }v'\} \qquad\text{if}\qquad v\ne v'. \end{align*} $$

If $d(v,v')=1$ we say v and $v'$ are adjacent. Also, for two subsets $\mathcal V_1,\mathcal V_2\subseteq \mathcal V$ , define

$$ \begin{align*} d(\mathcal V_1,\mathcal V_2) = \min\{d(v_1,v_2):v_1\in \mathcal V_1, v_2\in \mathcal V_2\}. \end{align*} $$

Set $d(v,v')=\infty $ if v and $v'$ belong to distinct connected components.

Example 2.1.1 The following path $\rho =e_1\ldots e_5$ from v to $v'$ has signature $(1,-1,-1,1,-1)$ and contains the $3$ -cycle $e_2e_3e_4$ . The circles denote other arbitrary elements in $\mathcal V$ .

(2.1.1)

Note $d(v,v')\le 2$ since $a_1a_5$ is a path from v to $v'$ . The subpath $e_2e_3$ is decreasing, while $e_3e_4e_5$ is an alternating subpath. The path $e_1e_5$ is alternating while $e_3e_2$ is increasing, but they are not subpaths of $\rho $ . $\diamond $

Every path gives rise to a subgraph associated with the set

$$ \begin{align*} \mathcal V^\rho = \{v\in \mathcal V: v\in e\text{ for some } e\in \rho \}. \end{align*} $$

Given $v\in \mathcal V$ , set $\mathcal A_v=\{v'\in \mathcal V:d(v,v')=1\}$ ,

(2.1.2) $$ \begin{align} \mathcal A_v^1 = \{v'\in\mathcal A_v: (v',v)\in\mathcal A \}, \qquad\text{and}\qquad \mathcal A_v^{-1} = \{v'\in\mathcal A_v: (v,v')\in\mathcal A \}. \end{align} $$

The valence of v is defined as $\#\mathcal A_v$ . If this number is $0$ , we say v is an isolated vertex, if it is $1$ , we say v is monovalent, and if it is at least $3$ , we say v is multivalent. Set

$$ \begin{align*} \mathring{G} = \{v\in\mathcal V: \#\mathcal A_v>1\} \qquad\text{and}\qquad \partial G = G\setminus\mathring{G}. \end{align*} $$

Elements of $\partial G$ will be referred to as boundary vertices while those of $\mathring {G}$ will be referred to as inner vertices. A vertex v is said to be a source if there are no incoming arrows toward it or, equivalently,

$$ \begin{align*} \mathcal A_v \subseteq \mathcal A_v^{-1}, \end{align*} $$

whereas it is a sink if

$$ \begin{align*} \mathcal A_v \subseteq \mathcal A_v^1. \end{align*} $$

In particular, isolated vertices are sinks and sources at the same time and a non-isolated vertex cannot be a sink and a source concomitantly. We will say a vertex is extremal if it is either a sink or a source. Note the middle circle in (2.1.1) is a source, the upper one is a sink, and the lower one is neither.

2.2 Cuts and special kinds of graphs

A cut of a directed graph G is a pair of subgraphs $(G',G")$ such that

$$ \begin{align*} \mathcal V = \mathcal V'\sqcup\mathcal V", \ \ \mathcal A'=\{a\in\mathcal A: h_a,t_a\in\mathcal V'\}, \ \ \text{and}\ \ \mathcal A"=\{a\in\mathcal A: h_a,t_a\in\mathcal V"\}. \end{align*} $$

The set

$$ \begin{align*} \mathcal A\setminus (\mathcal A'\cup\mathcal A") \end{align*} $$

is called the associated cut-set. Note the cut can be recovered from its cut-set if G is connected. Elements of the cut-set are said to cross the cut. An element $a\in \mathcal A$ is said to be a bridge if the number of connected components of $(\mathcal V,\mathcal A\setminus \{a\})$ is larger than that of G. If G is connected, this is equivalent to saying that $\{a\}$ is the cut-set of a cut. We shall say a cut $(G',G")$ is connected if both $G'$ and $G"$ are connected.

A connected graph with no cycles is said to be a tree. We shall refer to a tree with no multivalent vertex as a line. We will say G is a monotonic line if $\mathcal V=\mathcal V^\rho $ for some simple monotonic path $\rho $ . Note every tree with more than one vertex has at least two monovalent vertices, a fact which is false in general, as seen in the following examples.

(2.2.1)

Note that, in the first two graphs, the subgraphs obtained by removing the upper vertex are formed by directed cycles, while the cycle corresponding to the subgraphs obtained by removing the lower vertex are not. Note also that an arrow a is a bridge if and only if it is not contained in a cycle. In particular, the above graphs are bridgeless. A forest is a graph whose connected components are trees or, equivalently, every arrow is a bridge. We also recall that a tournament is a graph whose underlying set of edges is complete, i.e., $\{v,v'\}\in \mathcal E$ for every $v,v'\in \mathcal V, v\ne v'$ . In that case, the underlying nondirected graph is said to be complete. None of the above graphs is a tournament, but the middle one is missing only one arrow to become a tournament.

Let us record some elementary properties of trees.

In light of (iii) of the above lemma, given vertices $v,v'$ in a tree, we denote by $[v,v']$ the set of vertices of the unique element of $\mathscr P_{v,v'}$ . In particular, $[v',v]=[v,v']$ . Evidently, if $v\ne v'$ , $\#([v,v']\cap \mathcal A_v)= 1$ . Given $m\in \mathbb Z_{> 0}$ , set

(2.2.2) $$ \begin{align} \mathcal A_v^{\pm m}=\{v'\in\mathcal V: d(v,v')=m \text{ and } [v,v']\cap\mathcal A_{v'}^{\mp 1} \neq\emptyset\}. \end{align} $$

This clearly coincides with the sets defined in (2.1.2) when $m=1$ . Set also $\mathcal A_v^0=\{v\}$ and

(2.2.3) $$ \begin{align} \mathcal A_v^\pm = \bigcup_{m\in\mathbb Z_{>0}} \mathcal A_v^{\pm m}. \end{align} $$

We will be interested in graphs with no oriented cycles. In that case, the set of arrows $\mathcal A$ induces a partial order on $\mathcal V$ by the transitive extension of the strict relation

$$ \begin{align*} h_a\prec t_a \quad\text{for}\quad a\in\mathcal A. \end{align*} $$

Note

(2.2.4) $$ \begin{align} \mathcal P_{v,v'}^+ \ne\emptyset \ \Leftrightarrow\ v \prec v' \quad\text{and}\quad \mathcal P_{v,v'}^- \ne\emptyset \ \Leftrightarrow\ v' \prec v. \end{align} $$

Set $D(v,v')=0$ if $v=v'$ ,

(2.2.5) $$ \begin{align} \begin{aligned} D(v,v') &= \min\{\ell(\rho): \rho\in\mathscr{P}^+_{v,v'}\} \ \text{if}\ v \prec v',\\ D(v,v') &= -\min\{\ell(\rho): \rho\in\mathscr{P}^-_{v,v'}\} \ \text{if}\ v' \prec v, \end{aligned} \end{align} $$

and $D(v,v')=\infty $ if v and $v'$ are not comparable by $\preceq $ . Given $m\in \mathbb Z$ , set

(2.2.6) $$ \begin{align} \mathcal N^m_G(v) = \{v\in\mathcal V: D(v,v')=m\} \quad\text{and}\quad \mathcal N^\pm_G(v) = \bigcup_{m\in\mathbb Z_{\ge 0}} \mathcal N_G(v)^{\pm m}. \end{align} $$

If no confusion arises, we simplify notation and write $\mathcal N^m(v)$ and $\mathcal N^\pm (v)$ .

We shall say G is a totally ordered graph if $\preceq $ is a total order on $\mathcal V$ . The following lemma is easily established.

Only the last graph in (2.2.1) does not contain an directed cycle so $\preceq $ is defined, but it is not totally ordered. The following are examples of totally ordered graphs:

2.3 Classical and quantum algebras

Let I be the set of nodes of a finite-type connected Dynkin diagram. By regarding I as the set of vertices of the undirected graph whose edges are the sets of adjacent nodes of the diagram, we can use the notions of graph theory from the previous sections. By abuse of language, we refer to any subset J of I as a subdiagram (subgraph). In particular, we have defined $d(i,j)$ and $[i,j]$ for all $i,j\in I$ as well as $\partial J$ and $\mathring {J}$ for any $J\subseteq I$ . Let also $\bar J$ be the minimal connected subdiagram of I containing J. This is well defined since I is a tree.

Let $\mathfrak {g}$ be the simple Lie algebra over $\mathbb C$ corresponding to the given Dynkin diagram, fix a Cartan subalgebra $\mathfrak {h}$ and a set of positive roots $R^+$ and let $\mathfrak {g}_{\pm \alpha },\alpha \in R^+$ , and be the associated root spaces and triangular decomposition. The simple roots will be denoted by $\alpha _i$ , the fundamental weights by $\omega _i$ , $i\in I$ , while $Q,P,Q^+,P^+$ will denote the root and weight lattices with corresponding positive cones, respectively. Let also $h_\alpha \in \mathfrak {h}$ be the co-root associated with $\alpha \in R^+$ . If $\alpha =\alpha _i$ is simple, we often simplify notation and write $h_i$ . Let $C = (c_{i,j})_{i,j\in I}$ be the Cartan matrix of $\mathfrak {g}$ , i.e., $c_{i,j}=\alpha _j(h_i)$ , and $d_i, i\in I$ , be such that $d_ic_{i,j}=d_jc_{j,i}, i,j\in I$ . The Weyl group is denoted by $\mathcal {W}$ and its longest element by $w_0$ . We also denote by $w_0$ the involution on I induced by $w_0$ and set $i^*=w_0(i)$ . The dual Coxeter number and the lacing number of $\mathfrak {g}$ will be denoted by $h^\vee $ and $r^\vee $ , respectively. In particular, $r^\vee =\max \{d_i:i\in I\}$ .

For a subdiagram $J\subseteq I$ , let $\mathfrak {g}_J$ be the subalgebra of $\mathfrak {g}$ generated by the corresponding simple root vectors, $\mathfrak {h}_J=\mathfrak {h}\cap \mathfrak {g}_J$ , and so on. Let also $Q_J$ be the subgroup of Q generated by $\alpha _j, j\in J$ , $Q^+_J=Q^+\cap Q_J$ , and $R^+_J=R^+\cap Q_J$ . Given $\lambda \in P$ , let $\lambda _J$ denote the restriction of $\lambda $ to $\mathfrak {h}_J^*$ . For $\mu \in P$ , define also

$$\begin{align*}\mathrm{supp}(\mu)=\{i\in I:\mu(h_i)\ne 0\}.\\[-18pt]\end{align*}$$

For a Lie algebra $\mathfrak {a}$ over $\mathbb C$ , let $\tilde {\mathfrak {a}}=\mathfrak {a}\otimes \mathbb C[t,t^{-1}]$ be its loop algebras and identify $\mathfrak {a}$ with the subalgebra $\mathfrak {a}\otimes 1$ . Then, and $\tilde {\mathfrak {h}}$ is an abelian subalgebra.

Let $\mathbb F$ be an algebraically closed field of characteristic zero, fix $q\in \mathbb F^\times $ which is not a root of $1$ , and set $q_i=q^{d_i}, i\in I$ . Let also $U_q(\mathfrak {g})$ and $U_q(\tilde {\mathfrak {g}})$ be the associated Drinfeld–Jimbo quantum groups over $\mathbb F$ . We use the notation as in [Reference Moura26, Section 1.2]. In particular, the Drinfeld loop-like generators of $U_q(\tilde {\mathfrak {g}})$ are denoted by $x_{i,r}^\pm , h_{i,s}, k_i^{\pm 1}, i\in I, r,s\in \mathbb Z, s\ne 0$ . Also, $U_q(\mathfrak {g})$ is the subalgebra of $U_q(\tilde {\mathfrak {g}})$ generated by $x_i^\pm = x_{i,0}^\pm , k_i^{\pm 1}, i\in I$ , and the subalgebras $U_q(\mathfrak {n}^\pm ), U_q(\mathfrak {h}), U_q(\tilde {\mathfrak {n}}^\pm ), U_q(\tilde {\mathfrak {h}})$ are defined in the expected way.

Given $J\subseteq I$ , let $U_q(\mathfrak {a}_J)$ , with $\mathfrak {a}=\mathfrak {g}, \tilde {\mathfrak {g}},\tilde {\mathfrak {h}}$ , etc. be the respective quantum groups associated with $\mathfrak {a}_J$ . Let also $U_q(\mathfrak {a})_J$ be the subalgebra of $U_q(\tilde {\mathfrak {g}})$ generated by the generators corresponding to J. It is well known that there is an algebra isomorphism

$$ \begin{align*} U_q(\mathfrak{a})_J\cong U_{q_J}(\mathfrak{a}_J), \qquad\text{where}\qquad q_J=q^{d_J} \qquad\text{with}\qquad d_J = \min\{d_j:j\in J\}.\\[-18pt] \end{align*} $$

This is a Hopf algebra isomorphism only if $\mathfrak {a}\subseteq \mathfrak {g}$ . We shall always implicitly identify $U_q(\mathfrak {a})_J$ with $U_{q_J}(\mathfrak {a}_J)$ without further notice. When $J=\{j\}$ is a singleton, we simply write $U_q(\mathfrak {a})_j$ instead of $U_q(\mathfrak {a})_{\{j\}}$ , and so on.

2.4 The $\ell $ -weight lattice

The $\ell $ -weight lattice of $U_q(\tilde {\mathfrak {g}})$ is the multiplicative group $\mathcal P$ of n-tuples of rational functions $\boldsymbol {\varpi } = (\boldsymbol {\varpi }_i(u))_{i\in I}$ with values in $\mathbb F$ such that $\boldsymbol {\varpi }_i(0)=1$ for all $i\in I$ . The elements of the submonoid $\mathcal P^+$ of $\mathcal P$ consisting of n-tuples of polynomials will be referred to as dominant $\ell $ -weights or Drinfeld polynomials. If $\boldsymbol {\pi },\boldsymbol {\omega }\in \mathcal P^+$ satisfy $\boldsymbol {\pi }\boldsymbol {\omega }^{-1}\in \mathcal P^+$ , we shall say $\boldsymbol {\omega }$ divides $\boldsymbol {\pi }$ and write $\boldsymbol {\omega }|\boldsymbol {\pi }$ .

Given $a\in \mathbb F^\times $ and $\mu \in P$ , let $\boldsymbol {\omega }_{\mu ,a}\in \mathcal P$ be the element whose ith rational function is

$$ \begin{align*} (1-au)^{\mu(h_i)}, \quad i\in I. \end{align*} $$

In the case that $\mu =\omega _i$ for some i, we simplify notation and write $\boldsymbol {\omega }_{i,a}$ . Since $\mathcal P$ is a (multiplicative) free abelian group on the set $\{\boldsymbol {\omega }_{i,a}:i\in I,a\in \mathbb F^\times \}$ , there exists a unique group homomorphism $\mathrm {wt}:\mathcal P \to P$ determined by setting $\mathrm {wt}(\boldsymbol {\omega }_{i,a})=\omega _i$ . Set

$$ \begin{align*} \mathrm{supp}(\boldsymbol{\varpi}) = \mathrm{supp}(\mathrm{ wt}(\boldsymbol{\varpi})), \quad\boldsymbol{\varpi}\in\mathcal P. \end{align*} $$

There exists an injective map $\mathcal P\to (U_q(\tilde {\mathfrak {h}}))^*$ (see [Reference Moura26]) and, hence, we identify $\mathcal P$ with its image in $(U_q(\tilde {\mathfrak {h}}))^*$ .

Given $i\in I, a\in \mathbb F^\times , m\in \mathbb Z_{\ge 0}$ , define $q_i=q^{d_i}$ and

$$ \begin{align*} \boldsymbol{\omega}_{i,a,r} = \prod_{p=0}^{r-1} \boldsymbol{\omega}_{i,aq_i^{r-1-2p}}. \end{align*} $$

Note that $\mathrm {wt}(\boldsymbol {\omega }_{i,a,r}) = r\omega _i$ . We shall refer to Drinfeld polynomials of the form $\boldsymbol {\omega }_{i,a,r}$ as polynomials of KR type. Every Drinfeld polynomial can be written uniquely as a product of KR type polynomials such that, for every two factors supported at i, say $\boldsymbol {\omega }_{i,a,r}$ and $\boldsymbol {\omega }_{i,b,s}$ , the following holds:

(2.4.1) $$ \begin{align} \frac{a}{b} \ne q_i^{r+s-2p} \qquad\text{for all}\qquad 0\le p<\min\{r,s\}. \end{align} $$

Such factorization is said to be the q-factorization of $\boldsymbol {\pi }$ and the corresponding factors are called the q-factors of $\boldsymbol {\pi }$ . By abuse of language, whenever we mention the set of q-factors of $\boldsymbol {\pi }$ we actually mean the associated multiset of q-factors counted with multiplicities in the q-factorization. We shall say that $\boldsymbol {\pi },\boldsymbol {\pi }'\in \mathcal P^+$ have dissociate q-factorizations if the set of q-factors of $\boldsymbol {\pi }\boldsymbol {\pi }'$ is the union of the sets of q-factors of $\boldsymbol {\pi }$ and $\boldsymbol {\pi }'$ . It will also be convenient to work with factorizations in KR type polynomials which not necessarily satisfy (2.4.1). Such factorization will be referred to as pseudo q-factorizations and the associated factors as the corresponding pseudo q-factors.

For $\boldsymbol {\varpi }\in \mathcal P$ and $J\subseteq I$ , let $\boldsymbol {\varpi }_J$ be the associated J-tuple of rational functions, and let $\mathcal {P}_J=\{\boldsymbol {\varpi }_J:\boldsymbol {\varpi }\in \mathcal P\}$ . Similarly define $\mathcal {P}_J^+$ . Notice that $\boldsymbol {\varpi }_J$ can be regarded as an element of the $\ell $ -weight lattice of $U_q(\tilde {\mathfrak {g}})_J$ . Let $\pi _J:\mathcal P\to \mathcal {P}_J$ denote the map $\boldsymbol {\varpi }\mapsto \boldsymbol {\varpi }_J$ . If $J=\{j\}$ is a singleton, we write $\pi _j$ instead of $\pi _J$ .

Given $i\in I, a\in \mathbb F^\times $ , the following elements are known as simple $\ell $ -roots:

(2.4.2) $$ \begin{align} \boldsymbol{\alpha}_{i,a} = (\boldsymbol{\omega}_{i,aq_i,2})^{-1}\prod_{j\ne i} \boldsymbol{\omega}_{j,aq_i,-c_{j,i}}. \end{align} $$

The subgroup of $\mathcal P$ generated by them is called the $\ell $ -root lattice of $U_q(\tilde {\mathfrak {g}})$ and will be denoted by $\mathcal Q_q$ . Let also $\mathcal {Q}_q^+$ be the submonoid generated by the simple $\ell $ -roots. Quite clearly, $\mathrm {wt}(\boldsymbol {\alpha }_{i,a})=\alpha _i$ . Define a partial order on $\mathcal P$ by

$$\begin{align*}\boldsymbol{\varpi}\le\boldsymbol{\omega} \quad{if}\quad \boldsymbol{\omega}\boldsymbol{\varpi}^{-1}\in\mathcal{Q}_q^+.\end{align*}$$

2.5 Pre-factorization graphs

Given a set I, an I-coloring of a graph $G=(\mathcal V,\mathcal A)$ is a function $c:\mathcal V\to I$ . Given an I-coloring c and $i\in I$ , let $\mathcal V_i=\{x\in \mathcal V:c(x)=i\}$ . By a colored graph, we will mean an oriented graph G with a choice of coloring $c:\mathcal V\to I$ .

We shall also decorate the vertices and arrows of graphs by positive integers. We will refer to a function $\lambda :\mathcal V\to \mathbb Z_{>0}$ as a weight and to a function $\epsilon :\mathcal A\to \mathbb Z_{>0}$ as an exponent on G. The number $\epsilon (a)$ will be referred to as the exponent of a. We will always assume that $\epsilon $ satisfies the following compatibility condition:

(2.5.1) $$ \begin{align} \epsilon_\rho = \epsilon_{\rho'} \qquad\text{for all}\qquad \rho,\rho'\in\mathscr P_{v,v'}, v,v'\in \mathcal V, \end{align} $$

where, if $\rho =e_1\ldots e_m$ is such that $\sigma _\rho =(s_1,\dots ,s_m)$ and $e_j=\pi (a_j)$ ,

$$ \begin{align*} \epsilon_\rho := \sum_{j=1}^m s_j\epsilon(a_j). \end{align*} $$

Evidently, $\epsilon _{\rho ^-}=-\epsilon _{\rho }$ and one easily checks $\epsilon _{\rho *\rho '} = \epsilon _{\rho } + \epsilon _{\rho '}$ . Set

(2.5.2) $$ \begin{align} \mathscr P^+_G = \{\rho\in\mathscr P_G:\epsilon_\rho>0\} \quad\text{and}\quad \mathscr P^-_G = \{\rho\in\mathscr P_G:\epsilon_\rho<0\}. \end{align} $$

We shall refer to the data $(G,c,\lambda ,\epsilon )$ formed by a colored oriented graph, a weight $\lambda $ , and an exponent $\epsilon $ on G as a pre-factorization graph. We shall abuse of language and simply say G is a pre-factorization graph. We locally illustrate the structures maps of a pre-factorization graph with the following picture:

where i and j are the colors at the corresponding vertices, r and s are their associated weights, and m is the exponent associated with the given arrow. The following is an obvious consequence of (2.5.1).

In particular, only the last graph in (2.2.1) can be equipped with a pre-factorization graph structure. If I is as in Section 2.3 and G is a connected pre-factorization graph, for each choice of $(v_0,a)\in \mathcal V\times \mathbb F^\times $ , we can associate a Drinfeld polynomial as follows. Define

(2.5.3) $$ \begin{align} a_{v_0}=a \quad\text{and}\quad a_{v}=aq^{\epsilon_\rho} \quad\text{if}\quad \rho\in\mathscr P_{v_0,v}. \end{align} $$

Condition (2.5.1) guarantees this is well defined. Then, define

(2.5.4) $$ \begin{align} \boldsymbol{\pi}_{G,v_0,a}=\prod_{v\in \mathcal V} \boldsymbol{\omega}_{c(v),a_v,\lambda(v)}. \end{align} $$

One can easily check that

(2.5.5) $$ \begin{align} \boldsymbol{\pi}_{G,v_0,a'q^{-\epsilon_\rho}} = \boldsymbol{\pi}_{G,v_0',a'} \qquad\text{for all}\qquad (v_0',a')\in \mathcal V\times\mathbb F^\times,\ \rho\in\mathscr P_{v_0,v_0'}. \end{align} $$

Therefore, up to a uniform modification on the centers of the factors in the right-hand side of (2.5.4), the definition is independent of the choice of $(v_0,a)$ . We will often write $\boldsymbol {\pi }_G$ to shorten notation when the knowledge of precise centers is not relevant.

Example 2.5.2 Assume $\mathfrak {g}$ is of type $A_2$ , so $I=\{1,2\}$ , and consider the following pre-factorization graph:

If we select the middle vertex to define $\boldsymbol {\pi }=\boldsymbol {\pi }_G$ , we get

$$ \begin{align*} \boldsymbol{\pi} = \boldsymbol{\omega}_{2,a,2}\ \boldsymbol{\omega}_{1,aq^3,2}\ \boldsymbol{\omega}_{1,aq^4}. \end{align*} $$

Note that, in this case, the factors in (2.5.4) are the q-factors of $\boldsymbol {\pi }$ . However, this may not be the case as the following trivial example shows:

In this case, if we choose the first vertex as the base for the definition, the factors in (2.5.4) are $\boldsymbol {\omega }_{1,a}$ and $\boldsymbol {\omega }_{1,aq^2}$ , which combine to form a single q-factor.

2.6 Hopf algebra facts

We recall some general facts about Hopf algebras (see, for instance, [Reference Etingof, Gelaki, Nikshychm and Ostrik16] and the references therein).

Given a Hopf algebra $\mathcal H$ over $\mathbb F$ , its category $\mathcal C$ of finite-dimensional representations is an abelian monoidal category and we denote the (right) dual of a module V by $V^*$ . More precisely, the action of $\mathcal H$ of $V^*$ is given by

(2.6.1) $$ \begin{align} (hf)(v) = f(S(h)v) \quad\text{for}\quad h\in\mathcal H, f\in V^*, v\in V. \end{align} $$

The evaluation map $V^*\otimes V\to \mathbb F$ is a module map, where $\mathbb F$ is regarded as the trivial module by using the counit map. Moreover if, $v_1,\dots , v_n$ is a basis of V and $f_1, \dots , f_n$ is the corresponding dual basis, there exists a unique homomorphism of modules

$$ \begin{align*} \mathbb F\to V\otimes V^*, \qquad 1\mapsto \sum_{i=1}^n v_i\otimes f_i, \end{align*} $$

called the coevaluation map. We denote the evaluation and coevaluation maps associated with a module V by $\operatorname {ev}_V$ and $\operatorname {coev}_V$ , respectively, or simply by $\mathrm {ev}$ and $\mathrm {coev}$ if no confusion arises. In particular,

$$ \begin{align*} \operatorname{Hom}_{\mathcal H}(\mathbb F, V\otimes V^*) \ne 0 \qquad\text{and}\qquad \operatorname{Hom}_{\mathcal H}(V^*\otimes V, \mathbb F)\ne 0. \end{align*} $$

If the antipode is invertible, the notion of left dual module is obtained by replacing S by $S^{-1}$ in (2.6.1). The left dual of V will be denoted by $^*V$ and we have

$$ \begin{align*} ^*(V^*)\cong (^*V)^* \cong V. \end{align*} $$

Given $\mathcal H$ -modules $V_1,V_2,V_3$ , we have

(2.6.2) $$ \begin{align} \begin{aligned} & \operatorname{Hom}_{\mathcal C}(V_1\otimes V_2, V_3)\cong \operatorname{Hom}_{\mathcal C}(V_1, V_3\otimes V_2^*), \\\ & \operatorname{Hom}_{\mathcal C}(V_1, V_2\otimes V_3)\cong \operatorname{Hom}_{\mathcal C}(V_2^*\otimes V_1, V_3), \end{aligned} \end{align} $$

and

(2.6.3) $$ \begin{align} (V_1\otimes V_2)^*\cong V_2^*\otimes V_1^*. \end{align} $$

For instance, an isomorphism for the first statement in (2.6.2) is given by

$$ \begin{align*} f\mapsto (f\otimes\operatorname{id}_{V_2^*})\circ(\operatorname{id}_{V_1}\otimes\operatorname{coev}_{V_2})\circ\gamma_{V_1} \end{align*} $$

and has the inverse

$$ \begin{align*} g\mapsto\gamma^{\prime}_{V_3}\circ(\operatorname{id}_{V_3}\otimes\operatorname{ev}_{V_2})\circ(g\otimes\operatorname{id}_{V_2}), \end{align*} $$

where $\gamma _V:V\rightarrow V\otimes \mathbb {F}$ and $\gamma ^{\prime }_V:V\otimes \mathbb {F}\rightarrow V$ are the canonical maps. Note also that every short exact sequence

$$ \begin{align*} 0\to V_1\to V_2\to V_3\to 0 \end{align*} $$

gives rise to another short exact sequence of the form

(2.6.4) $$ \begin{align} 0\to V_3^*\to V_2^*\to V_1^*\to 0. \end{align} $$

We shall use the following lemma in the same spirit as in [Reference Kang, Kashiwara, Kim and Oh24] (a proof can also be found in [Reference Silva34]).

Lemma 2.6.1 Let $V_1,V_2,V_3\in \mathcal C$ and suppose M is a submodule of $V_1\otimes V_2$ and N is a submodule of $V_2\otimes V_3$ such that

$$\begin{align*}M\otimes V_3\subseteq V_1\otimes N.\end{align*}$$

Then, there exists a submodule W of $V_2$ such that

$$ \begin{align*} M\subseteq V_1\otimes W\qquad\textrm{and}\qquad W\otimes V_3\subseteq N. \end{align*} $$

Similarly, if $V_1\otimes N\subseteq M\otimes V_3,$ there exists a submodule W of $V_2$ such that

$$ \begin{align*} N\subseteq W\otimes V_3\qquad\textrm{and}\qquad V_1\otimes W\subseteq M. \end{align*} $$

Lemma 2.6.2 Let $V_1,V_2,V_3, L_1,L_2$ be $\mathcal H$ -modules and assume $V_2$ is simple. If

$$ \begin{align*} \varphi_1:L_1\rightarrow V_1\otimes V_2\quad\textrm{and}\quad\varphi_2: V_2\otimes V_3\rightarrow L_2 \end{align*} $$

are nonzero homomorphisms, the composition

$$\begin{align*}L_1\otimes V_3 \xrightarrow{\varphi_1\otimes \operatorname{id}_{V_3}} V_1\otimes V_2\otimes V_3 \xrightarrow{\operatorname{id}_{V_1}\otimes \varphi_2} V_1\otimes L_2 \end{align*}$$

does not vanish. Similarly, if

$$ \begin{align*} \varphi_1:V_1\otimes V_2\rightarrow L_1\quad\textrm{and}\quad\varphi_2:L_2\rightarrow V_2\otimes V_3 \end{align*} $$

are nonzero homomorphisms, the composition

$$\begin{align*}V_1\otimes L_2\xrightarrow{\operatorname{id}_{V_1}\otimes\varphi_2} V_1\otimes V_2\otimes V_3 \xrightarrow{\varphi_1\otimes\operatorname{id}_{V_3}} L_1\otimes V_3\end{align*}$$

does not vanish.

Proof We will write down the details for the first claim only, as the second can be proved similarly. Assume

$$ \begin{align*} (\operatorname{id}_{V_1}\otimes\varphi_2)\circ(\varphi_1\otimes\operatorname{id}_{V_3})=0, \end{align*} $$

i.e.,

$$\begin{align*}\operatorname{Im}(\varphi_1)\otimes V_3=\operatorname{Im}(\varphi_1\otimes\operatorname{id}_{V_3})\subseteq\operatorname{Ker}(\operatorname{id}_{V_1}\otimes\varphi_2)=V_1\otimes\operatorname{Ker}(\varphi_2).\end{align*}$$

Lemma 2.6.1 implies there exists a submodule $W\subseteq V_2$ such that

$$ \begin{align*} \operatorname{Im}(\varphi_1)\subseteq V_1\otimes W\quad\textrm{and}\quad W\otimes V_3\subseteq\operatorname{Ker}(\varphi_2). \end{align*} $$

Since $V_2$ is simple, either $W=0$ or $W=V_2$ . If $W=0$ , then $\operatorname {Im}(\varphi _1)\subseteq V_1\otimes W=0$ , which is a contradiction, since $\varphi _1$ is nonzero. On the other hand, if $W=V_2$ , it follows that $V_2\otimes V_3=\operatorname {Ker}(\varphi _2)$ , yielding a contradiction, since $\varphi _2$ is nonzero.

3.1 Finite-dimensional representations

Let $\mathcal {C}$ be the category of all finite-dimensional (type- $1$ ) weight modules of $U_q(\mathfrak {g})$ . Thus, a finite-dimensional $U_q(\mathfrak {g})$ -module V is in $\mathcal {C}$ if

$$ \begin{align*} V=\bigoplus_{\mu\in P}^{} V_\mu, \qquad\text{where}\qquad V_\mu=\{v\in V: k_iv=q_i^{\mu(h_i)}v \text{ for all } i\in I\}. \end{align*} $$

The following theorem summarizes the basic facts about $\mathcal {C}$ .

If $J\subseteq I$ we shall denote by $L_q(\lambda _J)$ the simple $U_q(\mathfrak {g})_J$ -module of highest weight $\lambda _J$ . Since $\mathcal {C}$ is semisimple, it is easy to see that, if $\lambda \in P^+$ and $v\in L_q(\lambda )_\lambda $ is nonzero, then $U_q(\mathfrak {g})_Jv\cong L_q(\lambda _J)$ .

Let $\widetilde {\mathcal {C}}$ the category of all finite-dimensional $\ell $ -weight modules of $U_q(\tilde {\mathfrak {g}})$ . Thus, a finite-dimensional $U_q(\tilde {\mathfrak {g}})$ -module V is in $\widetilde {\mathcal {C}}$ if

$$\begin{align*}V=\bigoplus_{\boldsymbol{\varpi}\in\mathcal P}^{} V_{\boldsymbol{\varpi,}}\end{align*}$$

where

$$\begin{align*}v\in V_{\boldsymbol{\varpi}} \quad\Leftrightarrow\quad \exists\ k\gg 0 \quad\text{s.t.}\quad (\eta-\boldsymbol{\varpi}(\eta))^kv=0 \quad\text{for all}\quad \eta\in U_q(\tilde{\mathfrak{h}}).\end{align*}$$

$V_{\boldsymbol {\varpi }}$ is called the $\ell $ -weight space of V associated with $\boldsymbol {\varpi }$ . Note that if $V\in {\widetilde {\mathcal {C}}}$ , then $V\in \mathcal {C}$ and

$$ \begin{align*} V_\mu = \bigoplus_{\boldsymbol{\varpi}:\mathrm{ wt}(\boldsymbol{\varpi})=\mu}^{} V_{\boldsymbol{\varpi}}. \end{align*} $$

If $V\in {\widetilde {\mathcal {C}}}$ , the q-character of V is the following element of the group ring $\mathbb Z[\mathcal P]$ :

$$ \begin{align*} \mathrm{qch}(V) = \sum_{\boldsymbol{\varpi}\in\mathcal P} \dim(V_{\boldsymbol{\varpi}})\boldsymbol{\varpi}. \end{align*} $$

A nonzero vector $v\in V_{\boldsymbol {\varpi }}$ is said to be a highest- $\ell $ -weight vector if

$$\begin{align*}\eta v=\boldsymbol{\varpi}(\eta)v \quad\text{for every}\quad \eta\in U_q(\tilde{\mathfrak{h}}) \quad\text{and}\quad x_{i,r}^+v=0 \quad\text{for all}\quad i\in I, r\in\mathbb Z.\end{align*}$$

V is said to be a highest- $\ell $ -weight module if it is generated by a highest- $\ell $ -weight vector. Evidently, every highest- $\ell $ -weight module has a maximal proper submodule and, hence, a unique irreducible quotient. In particular, if two simple modules are highest- $\ell $ -weight, then they are isomorphic if and only if the highest $\ell $ -weights are the same. This is also equivalent to saying that they have the same q-character. The following was proved in [Reference Chari and Pressley11].

It follows that $\mathrm {qch}(V)$ completely determines the irreducible factors of V. We shall denote by $L_q(\boldsymbol {\pi })$ any representative of the isomorphism class of simple modules with highest $\ell $ -weight $\boldsymbol {\pi }$ . For $J\subseteq I$ , we shall denote by $L_q(\boldsymbol {\pi }_J)$ the simple $U_q(\tilde {\mathfrak {g}})_J$ -module of highest weight $\boldsymbol {\pi }_J$ .

If V is a highest- $\ell $ -weight module with highest- $\ell $ -weight vector v and $J\subset I$ , we let $V_J$ denote the $U_q(\tilde {\mathfrak {g}})_J$ -submodule of $L_q(\boldsymbol {\pi })$ generated by v. Evidently, if $\boldsymbol {\pi }$ is the highest- $\ell $ -weight of V, then $V_J$ is highest- $\ell $ -weight with highest $\ell $ -weight $\boldsymbol {\pi }_J$ . Moreover, we have the following well-known facts:

(3.1.1) $$ \begin{align} V_J = \bigoplus_{\eta\in Q_J^+} V_{\mathrm{wt}(\boldsymbol{\pi})-\eta} = \bigoplus_{\eta\in Q_J} V_{\mathrm{wt}(\boldsymbol{\pi})+\eta}. \end{align} $$

3.2 Tensor products and duality for $U_q(\tilde {\mathfrak {g}})$ -modules

It is well known that $U_q(\tilde {\mathfrak {g}})$ is a Hopf algebra with invertible antipode. For the proof of following proposition, see [Reference Chari6, Propositions 1.5 and 1.6] (part (b) has not been proved there, but the proof is similar to that of part (c)).

Given $\boldsymbol {\pi }\in \mathcal {P}^+$ , define $\boldsymbol {\pi }^{\tau _a}\in \mathcal {P}^+$ by $\boldsymbol {\pi }^{\tau _a}_i(u)=\boldsymbol {\pi }_i(au).$ One easily checks that the pullback $L_q(\boldsymbol {\omega })^{\tau _a}$ of $L_q(\boldsymbol {\pi })$ by $\tau _a$ satisfies

(3.2.1) $$ \begin{align} L_q(\boldsymbol{\pi})^{\tau_a}\cong L_q(\boldsymbol{\pi}^{\tau_a}). \end{align} $$

Define also $\boldsymbol {\pi }^\sigma ,\boldsymbol {\pi }^*\in \mathcal P^+$ by

(3.2.2) $$ \begin{align} \boldsymbol{\pi}_i^\sigma(u) = \boldsymbol{\pi}_{i^*}(u) \quad\text{for}\quad i\in I, \quad\text{and}\quad \boldsymbol{\pi}^* = (\boldsymbol{\pi}^\sigma)^{\tau_{q^{-r^\vee h^\vee}}} = (\boldsymbol{\pi}^{\tau_{q^{-r^\vee h^\vee}}})^\sigma. \end{align} $$

It is well known that

(3.2.3) $$ \begin{align} L_q(\boldsymbol{\pi})^* \cong L_q(\boldsymbol{\pi}^*). \end{align} $$

We denote by $V^\sigma $ and $V^\kappa $ the pull-back of V by $\sigma $ and $\kappa $ , respectively. In particular,

(3.2.4) $$ \begin{align} (V_1\otimes V_2)^{\tau_a}\cong V_1^{\tau_a}\otimes V_2^{\tau_a}, \quad (V_1\otimes V_2)^\sigma\cong V_1^\sigma\otimes V_2^\sigma, \quad\text{and}\quad (V_1\otimes V_2)^\kappa\cong V_2^\kappa\otimes V_1^\kappa. \end{align} $$

Also, for any short exact sequence

$$ \begin{align*} 0\to V_1\to V_2\to V_3\to 0, \end{align*} $$

we have short exact sequences

(3.2.5) $$ \begin{align} 0\to V_1^f\to V_2^f\to V_3^f\to 0 \qquad\text{with}\qquad f=\tau_a,\sigma,\kappa. \end{align} $$

Moreover, if $\boldsymbol {\pi }\in \mathcal {P}^+$ with $\boldsymbol {\pi }_i(u) = \prod _j (1-a_{i,j}u)$ , where $a_{i,j}\in \mathbb F$ , and $\boldsymbol {\pi }^-\in \mathcal {P}^+$ is defined by $\boldsymbol {\pi }^-_i(u) = \prod _j (1-a_{i,j}^{-1}u)$ , we have

(3.2.6) $$ \begin{align} L_q(\boldsymbol{\pi})^\sigma \cong L_q(\boldsymbol{\pi}^\sigma) \quad\text{and}\quad L_q(\boldsymbol{\pi})^\kappa \cong L_q(\boldsymbol{\pi}^\kappa), \quad\text{where}\quad \boldsymbol{\pi}^\kappa = (\boldsymbol{\pi}^-)^*. \end{align} $$

It was proved in [Reference Frenkel and Reshetikhin17] that

(3.2.7) $$ \begin{align} \mathrm{qch}(V\otimes W) = \mathrm{qch}(V)\mathrm{qch}(W). \end{align} $$

In particular, we have the following proposition.

Given a connected subdiagram J, since $U_q(\tilde {\mathfrak {g}})_J$ is not a sub-coalgebra of $U_q(\tilde {\mathfrak {g}})$ , if M and N are $U_q(\tilde {\mathfrak {g}})_J$ -submodules of $U_q(\tilde {\mathfrak {g}})$ -modules V and W, respectively, it is in general not true that $M\otimes N$ is a $U_q(\tilde {\mathfrak {g}})_J$ -submodule of $V\otimes W$ . Recalling that we have an algebra isomorphism $U_q(\tilde {\mathfrak {g}})_J\cong U_{q_J}(\tilde {\mathfrak {g}}_J)$ , we shall denote by $M\otimes _J N$ the $U_q(\tilde {\mathfrak {g}})_J$ -module obtained by using the coalgebra structure from $ U_{q_J}(\tilde {\mathfrak {g}}_J)$ . The next result describes a special situation on which $M\otimes N$ is a submodule isomorphic to $M\otimes _J N$ . Recall the notation defined in the paragraph preceding Lemma 3.1.3.

Proof As shown in the proof of Proposition 3.2.3, we have

(3.2.8)

Thus, if $V\otimes W$ is highest- $\ell $ -weight, any nonzero vector in $V_J\otimes W_J$ is a linear combination of vectors of the form $x_{i_1,r_1}^-\ldots x_{i_l,r_l}^-(v\otimes w)$ for some $l\ge 0, i_k\in I, r_k\in \mathbb Z, 1\le k\le l$ . But the weight of such vector is

$$ \begin{align*} \mathrm{wt}(\boldsymbol{\pi})+\mathrm{ wt}(\boldsymbol{\varpi})-\sum_{k=1}^l \alpha_{i_k} \end{align*} $$

and, hence, we must have $i_k\in J$ for all $1\le k\le l$ , which implies the first claim. The second claim follows from the first together with Lemma 3.1.3.

3.3 Simple prime modules and q-factors

A finite-dimensional $U_q(\tilde {\mathfrak {g}})$ -module V is said to be prime if it is not isomorphic to a tensor product of two nontrivial modules. Evidently, any finite-dimensional simple module can be written as a tensor product of (simple) prime modules. If a prime module P appears in some factorization of a simple module S, we shall say that P is a prime factor of S.

In particular, in light of (3.2.7), in order to understand the q-characters of the simple modules, it suffices to understand those of the simple prime modules. However, the only case, the classification of simple prime modules is completely understood is for $\mathfrak {g}=\mathfrak {sl}_2$ . In that case, the classification is given by the following theorem, proved in [Reference Chari and Pressley11].

Theorem 3.3.1 If $\mathfrak {g}=\mathfrak {sl}_2,\boldsymbol {\pi }\in \mathcal P^+,$ and the q-factors of $\boldsymbol {\pi }$ are $\boldsymbol {\pi }^{(j)}, 1\le j\le m$ , then

$$ \begin{align*} L_q(\boldsymbol{\pi})\cong L_q(\boldsymbol{\pi}^{(1)})\otimes\cdots\otimes L_q(\boldsymbol{\pi}^{(m)}). \end{align*} $$

Moreover, up to re-ordering, $L_q(\boldsymbol {\pi })$ has a unique factorization as tensor product of prime modules. In particular, $L_q(\boldsymbol {\pi })$ is prime if and only if it has a unique q-factor.

If $\boldsymbol {\pi }\in \mathcal P^+$ has a unique q-factor, the module $L_q(\boldsymbol {\pi })$ is called a KR module. It is well known (see [Reference Chari7, Reference Oh and Scrimschaw32] and the references therein) that, given $(i,r),(j,s)\in I\times \mathbb Z_{>0}$ , there exists a finite set $\mathscr R_{i,j}^{r,s} \subseteq \mathbb Z_{>0}$ such that

(3.3.1) $$ \begin{align} L_q(\boldsymbol{\omega}_{i,a,r})\otimes L_q(\boldsymbol{\omega}_{j,b,s}) \text{ is reducible}\qquad\Leftrightarrow\qquad \frac{a}{b} = q^m \text{ with } |m|\in \mathscr R_{i,j}^{r,s}. \end{align} $$

Moreover, in that case,

(3.3.2) $$ \begin{align} L_q(\boldsymbol{\omega}_{i,a,r})\otimes L_q(\boldsymbol{\omega}_{j,b,s}) \text{ is highest-}\ell\text{-weight}\qquad\Leftrightarrow\qquad m>0. \end{align} $$

It follows from Proposition 3.2.2 and (2.6.3) that

(3.3.3) $$ \begin{align} \mathscr R_{j,i}^{s,r} = \mathscr R_{i,j}^{r,s} = \mathscr R_{i^*,j^*}^{r,s}. \end{align} $$

Theorem 3.3.2 If $\mathfrak {g}$ is of type A and $i,j\in I, r,s\in \mathbb Z_{>0}$ , we have

$$ \begin{align*} \mathscr R_{i,j}^{r,s} = \{r+s+d(i,j)-2p: - d([i,j],\partial I)\le p<\min\{r,s\} \}. \end{align*} $$

The above was essentially proved in [Reference Chari7] and can be read off the results of [Reference Oh and Scrimschaw32], from where the description for other types can also be extracted (see also [Reference Jakelić and Moura23]).

Given a connected subdiagram J such that $[i,j]\subseteq J$ , let $\mathscr R_{i,j,J}^{r,s}$ be determined by

$$ \begin{align*} L_q((\boldsymbol{\omega}_{i,a,r})_J)\otimes L_q((\boldsymbol{\omega}_{j,b,s})_J) \text{ is reducible}\qquad\Leftrightarrow\qquad \frac{a}{b} = q^m \text{ with } |m|\in \mathscr R_{i,j,J}^{r,s}. \end{align*} $$

Note this is not the same set obtained by considering the corresponding module for the algebra $U_{q_J}(\tilde {\mathfrak {g}}_J)\cong U_q(\tilde {\mathfrak {g}})_J$ . Indeed, if we denote the latter by $\mathscr R_{i,j}^{r,s}[J]$ , we have

$$ \begin{align*} m\in \mathscr R_{i,j}^{r,s}[J] \quad\Leftrightarrow\quad d_Jm\in \mathscr R_{i,j,J}^{r,s}. \end{align*} $$

Note also that Corollary 3.2.4 implies

(3.3.4) $$ \begin{align} \mathscr R_{i,j,J}^{r,s}\subseteq \mathscr R_{i,j,K}^{r,s} \quad\text{if}\quad J\subseteq K. \end{align} $$

Finally, set

(3.3.5) $$ \begin{align} \mathscr R_{i}^{r,s} = \mathscr R_{i,i,\{i\}}^{r,s}. \end{align} $$

Proof If the q-factorizations are not dissociate, it follows from Theorem 3.3.1 that there exists $i\in I$ and q-factors $\boldsymbol {\omega }$ of $\boldsymbol {\pi }$ and $\boldsymbol {\omega }'$ of $\boldsymbol {\pi }'$ , both supported at i, such that $L_q(\boldsymbol {\omega })\otimes L_q(\boldsymbol {\omega }')$ is reducible. Moreover, writing $\boldsymbol {\pi } = \widetilde {\boldsymbol {\pi }}\boldsymbol {\omega }$ and $\boldsymbol {\pi }'= \widetilde {\boldsymbol {\pi }}'\boldsymbol {\omega }'$ , it follows that

$$ \begin{align*} L_q(\boldsymbol{\pi}_i)\otimes L_q(\boldsymbol{\pi}^{\prime}_i)\cong L_q(\widetilde{\boldsymbol{\pi}}_i)\otimes L_q(\boldsymbol{\omega}_i)\otimes L_q(\boldsymbol{\omega}^{\prime}_i)\otimes L_q(\widetilde{\boldsymbol{\pi}}^{\prime}_i) \end{align*} $$

which is reducible, yielding a contradiction with Corollary 3.2.4.

Given $\boldsymbol {\pi }\in \mathcal P^+$ , consider a nontrivial $2$ -set partition of its set of q-factors and let $\boldsymbol {\omega }$ and $\boldsymbol {\varpi }$ be the products of the q-factors in each of the parts. The above corollary tells us that the task of deciding the primality of $L_q(\boldsymbol {\pi })$ can be phrased as a task of testing the reducibility of $L_q(\boldsymbol {\omega })\otimes L_q(\boldsymbol {\varpi })$ for every such partition. Thus, one can think of organizing the level of complexity of the task by the number of q-factors of $\boldsymbol {\pi }$ . The answer for the two first levels is given by:

3.4 Factorization graphs

Recall the definition of the sets $\mathscr R_{i,j}^{r,s}$ in (3.3.1), as well as (3.3.5), and (2.5.2). We shall say that a pre-factorization graph G is a q-factorization graph if, for every $i\in I$ ,

(3.4.1) $$ \begin{align} v,v'\in \mathcal V_i, \ \rho\in\mathscr P_{v,v'} \qquad\Rightarrow\qquad |\epsilon_\rho| \notin \mathscr R_i^{\lambda(v),\lambda(v')} \end{align} $$

and

(3.4.2) $$ \begin{align} \rho\in\mathscr P_{v,v'}\cap\mathscr P_G^+ \quad\text{with}\quad \epsilon_\rho \in \mathscr R_{c(v),c(v')}^{\lambda(v),\lambda(v')} \qquad\Rightarrow\qquad (v',v)\in\mathcal A. \end{align} $$

Condition (3.4.1) ensures that the factors in the right-hand side of (2.5.4) are the q-factors of $\boldsymbol {\pi }$ . On the other hand, (3.4.2) guarantees that no pre-factorization graph can be obtained by adding an arrow to G. We will refer to a pre-factorization graph satisfying (3.4.2) as a pseudo q-factorization graph.

We shall now see that any pseudo q-factorization of a Drinfeld polynomial gives rise to a pseudo q-factorization graph which is a q-factorization graph if and only if it is the q-factorization. Thus, fix a Drinfeld polynomial $\boldsymbol {\pi }$ , and let $\mathcal V$ be the corresponding multiset of pseudo q-factors. For $i\in I$ , let

$$ \begin{align*} \mathcal V_i = \{\boldsymbol{\omega}\in \mathcal V: \mathrm{ supp}(\boldsymbol{\omega})=\{i\}\}. \end{align*} $$

This gives rise to a coloring $c:\mathcal V\to I$ defined by declaring $\mathcal V_i=c^{-1}(\{i\})$ . The weight map $\lambda :\mathcal V\to \mathbb Z_{>0}$ is defined by

(3.4.3) $$ \begin{align} \lambda(\boldsymbol{\omega}) = \mathrm{wt}(\boldsymbol{\omega})(h_i) \qquad\text{for all}\qquad \boldsymbol{\omega}\in \mathcal V_i. \end{align} $$

In particular,

(3.4.4) $$ \begin{align} \sum_{i\in I}\sum_{\boldsymbol{\omega}\in \mathcal V_i} \lambda(\boldsymbol{\omega})\omega_i = \mathrm{wt}(\boldsymbol{\pi}). \end{align} $$

The set of arrows $\mathcal A=\mathcal A(\boldsymbol {\pi })$ is defined as the set of ordered pairs of q-factors, say $(\boldsymbol {\omega }_{i,a,r},\boldsymbol {\omega }_{j,b,s})$ , such that

(3.4.5) $$ \begin{align} a=bq^m \quad\text{for some}\quad m\in \mathscr R_{i,j}^{r,s}. \end{align} $$

In representation theoretic terms, this is equivalent to saying:

$$ \begin{align*} L_q(\boldsymbol{\omega}_{i,a,r})\otimes L_q(\boldsymbol{\omega}_{j,b,s}) \quad\text{is reducible and highest-}\ell\text{-weight.} \end{align*} $$

Note that, in the case of the actual q-factorization, we necessarily have $m\notin \mathscr R_i^{r,s}$ when $i=j$ . The value of the exponent $\epsilon :\mathcal A\to \mathbb Z_{>0}$ at an arrow satisfying (3.4.5) is set to be m. Quite clearly, $G= (\mathcal V,\mathcal A)$ with the above choice of coloring, weight, and exponent is a pseudo q-factorization graph and $\boldsymbol {\pi }_G =\boldsymbol {\pi }$ . We refer to G as a pseudo q-factorization graph over $\boldsymbol {\pi }$ . In the case this construction was performed using the q-factorization of $\boldsymbol {\pi }$ , then G will be called the q-factorization graph of $\boldsymbol {\pi }$ and we denote it by $G(\boldsymbol {\pi })$ .

It is now natural to seek for the classification of the prime q-factorization graphs, i.e., those for which $L_q(\boldsymbol {\pi }_G)$ is prime. In type $A_1$ , this is the case if and only if the q-factorization graph of $\boldsymbol {\pi }$ has a single vertex, which is also equivalent to saying that the graph is connected. For higher rank, the story is much more complicated. We still have the following proposition which will be proved in Section 4.1.

Proposition 3.4.1 Let $\boldsymbol {\pi }\in \mathcal P^+$ . If $G_1, \ldots , G_k$ are the connected components of $G(\boldsymbol {\pi })$ and $\boldsymbol {\pi }^{(j)}\in \mathcal P^+, 1\le j\le k$ , are such that $\boldsymbol {\pi } = \prod _{j=1}^k \boldsymbol {\pi }^{(j)}$ and $G_j = G(\boldsymbol {\pi }^{(j)})$ , then

$$ \begin{align*} L_q(\boldsymbol{\pi})\cong L_q(\boldsymbol{\pi}^{(1)})\otimes\cdots\otimes L_q(\boldsymbol{\pi}^{(k)}). \end{align*} $$

In particular, $G(\boldsymbol {\pi })$ is connected if $L_q(\boldsymbol {\pi })$ is prime.

However, even for type $A_2$ , the converse is not true and counter examples can be found in [Reference Moura and Silva27], for instance.

We also introduce duality notions for pre-factorization graphs. Given a graph G, we denote by $G^-$ the graph obtained from G by reversing all the arrowsFootnote ² and keeping the rest of structure of (pre)-factorization graph. In light of (3.3.3), $G^-$ is a factorization graph as well, which we refer to as the arrow-dual of G. Similarly, the graph $G^*$ , called the color-dual of G, obtained by changing the coloring according to the rule $i\mapsto i^*$ for all $i\in I$ , is a factorization graph. Moreover,

(3.4.6) $$ \begin{align} \boldsymbol{\pi}_{G^-,v,a^{-1}} =\boldsymbol{\pi}_{G,v,a}^- \qquad\text{and}\qquad \boldsymbol{\pi}_{G^*,v,aq^{-r^\vee h^\vee}} =\boldsymbol{\pi}_{G,v,a}^*. \end{align} $$

Given $\boldsymbol {\pi },\boldsymbol {\pi }'\in \mathcal P^+$ , the graph $G(\boldsymbol {\pi }\boldsymbol {\pi }')$ may have no relation to $G(\boldsymbol {\pi })$ and $G(\boldsymbol {\pi }')$ . However, if $\boldsymbol {\pi }$ and $\boldsymbol {\pi }'$ have dissociate q-factorizations, then $G(\boldsymbol {\pi })$ and $G(\boldsymbol {\pi }')$ determine a cut of $G(\boldsymbol {\pi }\boldsymbol {\pi }')$ . We will consider several times the situation that the corresponding cut-set is a singleton. More generally, given pseudo q-factorization graphs G and $G'$ , let $G\otimes G'$ be the unique pseudo q-factorization graph whose set of vertices is $\mathcal V\sqcup \mathcal V'$ preserving the original coloring and weight map. The following is trivially established.

If G and $G'$ are pseudo q-factorization graphs over $\boldsymbol {\pi }$ and $\boldsymbol {\pi }'$ , respectively, we shall say that the pair $(G,G')$ or, equivalently, that $G\otimes G'$ is simple if so is $L_q(\boldsymbol {\pi })\otimes L_q(\boldsymbol {\pi }')$ . Otherwise we say it is reducible. Evidently, $G\otimes G'=G'\otimes G$ , regardless if $L_q(\boldsymbol {\pi })\otimes L_q(\boldsymbol {\pi }')$ is simple or not.

3.5 Main conjectures and results

Throughout this section, we let $G=G(\boldsymbol {\pi })=(\mathcal V,\mathcal A)$ be a q-factorization graph. We say G is prime if $L_q(\boldsymbol {\pi })$ is prime. One could hope that the complexity of prime graphs is incremental in the sense that all prime graphs with $N+1$ vertices are obtained by adding a vertex in some particular ways to some prime graph with N vertices. In other words:

Trees are the simplest kinds of directed graphs and, among them, totally ordered lines are the simplest. The following is the first of our main results.

For $\mathfrak {g}$ of type $A_2$ , this was the main result of [Reference Chari and Pressley14]. It will be proved here for general $\mathfrak {g}$ , by a different argument, as a corollary of Proposition 4.3.1. In particular, differently than the proof in [Reference Chari and Pressley14], our proof does not use any information about the elements belonging to the sets $\mathscr R_{i,j}^{r,s}$ . For $\mathfrak {g}$ of type A, we also prove the following generalization, which is the main result of the present paper.

In particular, a q-factorization graph afforded by a tournament is prime. After Theorem 3.5.5, it becomes natural the purely combinatorial problem of classifying all q-factorization graphs which are totally ordered since this leads to the explicit construction of a family of Drinfeld polynomials whose corresponding simple modules are prime. We shall not pursue a general answer for this combinatorial problem here. However, for illustrative purposes, we do present two results in this direction. One is Example 3.6.1, where we describe an infinite family of examples of q-factorization graphs afforded by tournaments. The other is the following proposition whose proof, given in Section 5.2, is essentially a byproduct of some technical lemmas extracted from the proof of Theorem 3.5.5 in Section 5.1. In particular, it solves this combinatorial problem for type $A_2$ since, in that case, $I=\partial I$ .

We have the following rephrasing of Corollary 3.3.5 in the language of cuts: G is prime if and only if every nontrivial cut of G is reducible. Several partial results in the direction of proving our main results provide criteria for checking the reducibility of certain special cuts, which we deem to be interesting results in their own right. One of these, which is an immediate consequence of Corollary 4.3.3, is stated here in the language of cuts.

The following corollary about triangles is immediate.

This corollary implies there is only one cut for a triangle which may be simple: the one whose singleton contains the vertex which is not extremal. Theorem 3.5.5 implies this is not so for type A. However, the present proof utilizes the precise description of the sets $\mathscr R_{i,j}^{r,s}$ and, hence, in order to extend it to other types, it requires a case by case analysis, which will appear elsewhere.

We also have the following general criterion for primality.

Proposition 3.5.9 will be proved as a corollary to Proposition 4.5.3.

3.6 Examples

The first example provides a family of tournaments for type A and, hence, a family of simple prime modules.

Example 3.6.1 Given $N>1$ , let $\mathfrak {g}$ be of type $A_n, n\ge {3N-4}$ , identify I with the integer interval $[1,n]$ as usual, and consider

$$ \begin{align*} \boldsymbol{\pi} = \prod\limits_{i=1}^{N}\boldsymbol{\omega}_{i+N-2, q^{3(i-1)}}. \end{align*} $$

Checking that $G(\boldsymbol {\pi })$ is a tournament with N vertices amounts to showing that

$$ \begin{align*} 3(j-i)\in\mathscr R_{i+N-2, j+N-2}^{1,1} \quad\text{for all}\quad 1\le i<j\le N, \end{align*} $$

which, by Theorem 3.3.2, is equivalent to

$$ \begin{align*} 3(j-i) = 2 + d(i+N-2, j+N-2) -2p \quad\text{with}\quad -d([i+N-2,j+N-2], \partial I)\le p\le 0. \end{align*} $$

Since $d(i+N-2, j+N-2)=j-i$ and $3(j-i)=2 + (j-i) -2(1-(j-i))$ , we need to check

$$ \begin{align*} -d([i+N-2,j+N-2], \partial I)\le 1-(j-i)\le 0. \end{align*} $$

The second inequality is immediate from $1\leq i<j\leq N$ . On the other hand,

$$ \begin{align*} d([i+N-2,j+N-2], \partial I) = \min\{ (i+N-2)-1, n-(j+N-2) \}\ge N-2, \end{align*} $$

from where the first inequality easily follows.

Although the above family affords triangles only for rank at least $5$ , it is easy to build q-factorization graphs which are triangles for $n\ge 3$ . For instance,

Example 3.6.2 We now examine prime snake modules for type A from the perspective of q-factorization graphs. The notion of snake and snake modules was introduced in [Reference Mukhin and Young28] while that of prime snakes was introduced in [Reference Mukhin and Young29] and revised in [Reference Duan, Li and Luo15]. We now rephrase these definitions in terms of the sets $\mathscr R_{i,j}^{r,s}$ . A (type A) snake of length k is a sequence $(i_j,m_j)\in \mathcal I\times \mathbb Z, 1\le j\le k$ , such that

$$ \begin{align*} m_{j+1}-m_j = 2+d(i_j,i_{j+1})-2p_j \quad\text{for some}\quad p_j\in\mathbb Z_{\le 0} \quad\text{and all}\quad 1\le j<k. \end{align*} $$

The snake is said to be prime if $-d([i_j,i_{j+1}],\partial I)\le p_j$ for all $1\le j<k$ . In other words, the snake is prime if and only if

$$ \begin{align*} m_{j+1}-m_j \in \mathscr R_{i_j,i_{j+1}}^{1,1} \quad\text{for all}\quad 1\le j<k. \end{align*} $$

Given a snake and $a\in \mathbb F^\times $ , the associated snake module is $L_q(\boldsymbol {\pi })$ with

(3.6.1) $$ \begin{align} \boldsymbol{\pi} = \prod_{j=1}^k \boldsymbol{\omega}_{i_j,aq^{m_j}}. \end{align} $$

For a general snake, $G(\boldsymbol {\pi })$ may be disconnected and, hence, not prime. However, if the snake is prime and we regard the definition of $\boldsymbol {\pi }$ as a pseudo q-factorization, the associated pseudo q-factorization graph is totally ordered. It is then easy to see that $G(\boldsymbol {\pi })$ , the actual q-factorization graph, is also totally ordered and, hence, prime by Theorem 3.5.5. Thus, Theorem 3.5.5, together with Proposition 3.4.1, recovers [Reference Mukhin and Young29, Proposition 3.1].

The following is the q-factorization graph arising from the following prime snake for type $A_5$ : $(4,-2), (3,1), (2,4), (3,7)$ .

Example 3.6.3 Snake modules also arise in the study of the so-called skew representations associated with skew tableaux $\lambda \backslash \mu $ [Reference Hopkins and Molev22, Section 4], which we now review. Fix $m\in \mathbb Z_{\ge 0}$ , as well as $\lambda = (\lambda _1,\lambda _2,\dots ,\lambda _{m+n+1})\in \mathbb Z^{m+n+1}$ and $\mu = (\mu _1,\dots ,\mu _m)\in \mathbb Z^m$ such that

$$ \begin{align*} \lambda_i\ge \lambda_{i+1}, \quad \mu_l\ge \mu_{l+1}, \quad\text{and}\quad \lambda_k\ge \mu_k \ge\lambda_{k+n+1} \end{align*} $$

for all $1\le i\le m+n, 1\le l<m, 1\le k\le m$ . Set $\mu _0 = +\infty , \mu _{m+1}= -\infty $ , and, for $1\le i\le n+1$ and $1\le l\le m+1$ , let $\nu _{i,l}$ be the middle value among $\mu _{l-1},\mu _l$ , and $\lambda _{i+l-1}$ . The skew module of $U_q(\tilde {\mathfrak {g}})$ associated with the skew tableaux $\lambda \setminus \mu $ is the simple module $L_q(\boldsymbol {\pi }^{\lambda ,\mu })$ , where

(3.6.2) $$ \begin{align} \boldsymbol{\pi}^{\lambda,\mu}_i(u) = \prod_{l=1}^{m+1} \boldsymbol{\omega}_{i,q^{\nu_{i,l}+\nu_{i+1,l}-2l+1-i},\nu_{i,l}-\nu_{i+1,l}}. \end{align} $$

For $\boldsymbol {\mu }=\emptyset $ , this is the Drinfeld polynomial of an evaluation module. With a little patience, one can check each factor of the above definition is a q-factor of $\boldsymbol {\pi }^{\lambda ,\mu }$ . Moreover, each connected component of $G(\boldsymbol {\pi }^{\lambda ,\mu })$ is totally ordered and, hence, corresponds to a prime simple module by Theorem 3.5.5.

In order to explore specific examples, let us organize the table:

Plugging this information in (3.6.2), each row will produce at most one q-factor for each $i\in I$ . Moreover, the centers of the associated q-strings are of the form $q^k$ for some exponent $k\in \mathbb Z$ . We can then form a table with the corresponding exponents and lengths.

For instance, if $\lambda = (20,16,10,7,2,0)$ and $\mu = (17,5)$ , so $m=2$ and $n=3$ , we have

Organizing the vertices of $G(\boldsymbol {\pi }^{\lambda ,\mu })$ following the rows of the last table, we get

Thus, this example leads to a graph with two connected components: a singleton and an oriented line. If $\lambda = (6,6,6,4,2,1,1)$ and $\mu = (5)$ , so $m=1$ and $n=5$ , then

and $G(\boldsymbol {\pi }^{\lambda ,\mu })$ is connected:

Although the underlying directed graph is the same as the one in Example 3.6.2, $L_q(\boldsymbol {\pi }^{\lambda ,\mu })$ is not a snake module. Indeed, $\boldsymbol {\pi }^{\lambda ,\mu }$ can be constructed as in (3.6.1) by using the sequence:

$$\begin{align*}(4,-4), (4,-2), (3,-1), (3,1), (2,4), (3,7).\end{align*}$$

However, this is not a snake because $(i_2,m_2)=(4,-2), (i_3,m_3) = (3,-1)$ , and $m_3-m_2 = 1 \notin \mathscr R_{4,3}^{1,1}$ . One can easily check no reordering of this sequence is a snake.

4.1 Background on highest- $\ell $ -weight tensor products

The following is easily established.

Proof Assume we have a monomorphism $L_q(\boldsymbol {\varpi })\to V$ . It follows from (2.6.4) and (2.6.3) that we have an epimorphism

$$ \begin{align*} L_q(\boldsymbol{\pi}')^*\otimes L_q(\boldsymbol{\pi})^* \to L_q(\boldsymbol{\varpi})^*. \end{align*} $$

In particular, letting $\psi = \sigma \circ \tau _{q^{r^\vee h^\vee }}$ and using (3.2.4) and (3.2.5), we get an epimorphism

$$ \begin{align*} (L_q(\boldsymbol{\pi}')^*)^\psi\otimes (L_q(\boldsymbol{\pi})^*)^\psi \to (L_q(\boldsymbol{\varpi})^*)^\psi. \end{align*} $$

One easily checks using (3.2.1), (3.2.3), and (3.2.6) that the domain of the latter epimorphism is isomorphic to W and $ (L_q(\boldsymbol {\varpi })^*)^\psi \cong L_q(\boldsymbol {\varpi })$ . The converse in part (a) is proved by reversing this argument. Part (b) is immediate from (a) since the assumption on W implies we have an epimorphism $W\to L_q(\boldsymbol {\pi }\boldsymbol {\pi }')$ and the top weight space of V is one-dimensional and equal to $V_{\boldsymbol {\pi }\boldsymbol {\pi }'}$ .

For proving (c), let $V'$ be the submodule of V generated by its top weight space. The assumption is equivalent to saying that $V'$ is a proper submodule. Since V is finite-dimensional, the set of proper submodules of V containing $V'$ is nonempty and contains a maximal element, say U, which is necessarily also maximal in the set of all proper submodules of V. Hence, $V/U\cong L_q(\boldsymbol {\varpi })$ for some $\boldsymbol {\varpi }\in \mathcal P^+$ and, since $V_{\boldsymbol {\pi }\boldsymbol {\pi }'}\subseteq U$ , we have $\boldsymbol {\varpi }\ne \boldsymbol {\pi }\boldsymbol {\pi }'$ . Since $\boldsymbol {\varpi }$ is an $\ell $ -weight of V and every $\ell $ -weight of V is smaller than $\boldsymbol {\pi }\boldsymbol {\pi }'$ , we have $\boldsymbol {\varpi }<\boldsymbol {\pi }\boldsymbol {\pi }'$ .

The following fact is well known (a proof can be found in [Reference Moura26]).

Proof If $U:=L_q(\boldsymbol {\pi })\otimes L_q(\boldsymbol {\varpi })$ is simple, Proposition 3.2.2 implies $U\cong W:=L_q(\boldsymbol {\varpi })\otimes L_q(\boldsymbol {\pi })$ . In particular, U and W are both highest- $\ell $ -weight. Conversely, assume U and W are both highest- $\ell $ -weight. Since U is highest- $\ell $ -weight, Proposition 4.1.3, (2.6.3), and (3.2.3), imply that it suffices to show

(4.1.1) $$ \begin{align} L_q(\boldsymbol{\varpi}^*)\otimes L_q(\boldsymbol{\pi}^*) \quad\text{is highest-}\ell\text{-weight.} \end{align} $$

Using (3.2.5) with $V_2=W$ , it follows from (3.2.4) and (3.2.6) that $W^\sigma \cong L_q(\boldsymbol {\varpi }^\sigma )\otimes L_q(\boldsymbol {\pi }^\sigma )$ is highest- $\ell $ -weight. Setting $a= q^{-r^\vee h^\vee }$ , (3.2.6) implies $\boldsymbol {\pi }^*=(\boldsymbol {\pi }^\sigma )^{\tau _a}$ and similarly for $\boldsymbol {\varpi }$ . The proof of (4.1.1) is then completed by using (3.2.4), (3.2.1) and (3.2.5) with $V_2 = W^\sigma $ and $f=\tau _a$ .

We now state one of the main tools we shall use in the proofs of our main results.

Proof The first claim, which is the part we need below, is the main result of [Reference Hernandez19] (see also Remark 4.1.7). We now prove the second by induction on m. Thus, suppose $S_1\otimes \cdots \otimes S_m$ is highest- $\ell $ -weight and note there is nothing to prove if $m\le 2$ . Assume $m>2$ and note Lemma 4.1.1 implies $S_i\otimes \cdots \otimes S_j$ is highest- $\ell $ -weight for all $1\le i\le j\le m$ . Together with the induction hypothesis, this implies $S_i\otimes S_j$ is highest- $\ell $ -weight for all $1\le i<j\le m$ except if $(i,j)=(1,m)$ .

To prove that $S_1\otimes S_m$ is also highest- $\ell $ -weight, thus completing the proof, let S be a simple quotient of $T:=S_2\otimes \cdots \otimes S_{m-1}$ and consider the associated epimorphism $\pi :T\to S$ . This implies we have an epimorphism

$$ \begin{align*} S_1\otimes T\otimes S_m\xrightarrow{\quad\operatorname{id}_{S_1}\otimes\pi\otimes\operatorname{id}_{S_m}\quad}S_1\otimes S\otimes S_m, \end{align*} $$

which, together with the assumption that $S_1\otimes \cdots \otimes S_m$ is highest- $\ell $ -weight, implies $S_1\otimes S\otimes S_m$ is highest- $\ell $ -weight as well. The inductive argument is completed if $m>3$ .

If $m=3$ , let $\lambda _i\in P^+$ be the highest weight of $S_i, 1\le i\le 3$ . Since $S_1\otimes S_2$ is highest- $\ell $ -weight, Lemma 4.1.2(b) implies the submodule M generated by the top weight space of $S_2\otimes S_1$ is simple and we have a monomorphism

$$ \begin{align*} M\otimes S_3 \to S_2\otimes S_1\otimes S_3. \end{align*} $$

If $S_1\otimes S_3$ were not highest- $\ell $ -weight, an application of Lemma 4.1.2(c) would gives us an epimorphism

$$ \begin{align*} S_1\otimes S_3 \to N, \end{align*} $$

where N is a simple module whose highest weight $\lambda $ satisfies $\lambda <\lambda _1+\lambda _3$ . Lemma 2.6.2 says these maps can be used to obtain a nonzero map

$$ \begin{align*} M\otimes S_3 \to S_2\otimes N. \end{align*} $$

The highest weight of $M\otimes S_3$ is $\lambda _1+\lambda _2+\lambda _3$ while that of $S_2\otimes N$ is $\lambda _2+\lambda $ . To reach a contradiction, it then suffices to show $M\otimes S_3$ is highest- $\ell $ -weight.

Indeed, we have an epimorphism $S_1\otimes S_2\to M$ and, hence, an epimorphism

$$ \begin{align*} S_1\otimes S_2\otimes S_3 \to M\otimes S_3. \end{align*} $$

The assumption that $S_1\otimes S_2\otimes S_3$ is highest- $\ell $ -weight then implies that so is $M\otimes S_3$ , as desired.

Corollary 4.1.6 Given $\boldsymbol {\pi },\widetilde {\boldsymbol {\pi }}\in \mathcal {P}^+$ , $L_q(\boldsymbol {\pi })\otimes L_q(\widetilde {\boldsymbol {\pi }})$ is highest- $\ell $ -weight if there exist $\boldsymbol {\pi }^{(k)}\in \mathcal P^+, 1\le k\le m$ , $\widetilde {\boldsymbol {\pi }}^{(k)}\in \mathcal P^+, 1\le k\le \tilde m$ , such that

$$ \begin{align*} \boldsymbol{\pi} = \prod_{k=1}^m \boldsymbol{\pi}^{(k)}, \quad \widetilde{\boldsymbol{\pi}} = \prod_{k=1}^{\tilde m} \widetilde{\boldsymbol{\pi}}^{(k)}, \end{align*} $$

and the following tensor products are highest- $\ell $ -weight:

$$ \begin{gather*} L_q(\boldsymbol{\pi}^{(k)})\otimes L_q(\boldsymbol{\pi}^{(l)}), \quad L_q(\widetilde{\boldsymbol{\pi}}^{(k)})\otimes L_q(\widetilde{\boldsymbol{\pi}}^{(l)}), \quad\text{for } k\le l,\\ \text{ and}\quad L_q(\boldsymbol{\pi}^{(k)})\otimes L_q(\widetilde{\boldsymbol{\pi}}^{(l)}) \quad\text{for all }k,l. \end{gather*} $$

Moreover, if all these tensor products are irreducible, then so is $L_q(\boldsymbol {\pi })\otimes L_q(\widetilde {\boldsymbol {\pi }})$ .

Proof It follows from Theorem 4.1.5 that

$$ \begin{align*} W:= L_q(\boldsymbol{\pi}^{(1)})\otimes \cdots\otimes L_q(\boldsymbol{\pi}^{(m)}), \quad \tilde W:= L_q(\widetilde{\boldsymbol{\pi}}^{(1)})\otimes \cdots\otimes L_q(\widetilde{\boldsymbol{\pi}}^{(\tilde m)}), \quad\text{and}\quad W\otimes \tilde W \end{align*} $$

are highest- $\ell $ -weight. Therefore, we have epimorphisms $p:W\to L_q(\boldsymbol {\pi }), \tilde p:\tilde W\to L_q(\widetilde {\boldsymbol {\pi }})$ , and, hence, $p\otimes \tilde p:W\otimes \tilde W\to L_q(\boldsymbol {\pi })\otimes L_q(\widetilde {\boldsymbol {\pi }})$ . For the last claim, the extra assumption implies we reach the same conclusion with the tensor products above in reversed order. Hence, we are done by Corollary 4.1.4.

We are now able to prove Proposition 3.4.1.

4.2 On the removal of boundary vertices

Lemma 4.2.1 Let $\boldsymbol {\pi }\in \mathcal P^+$ and suppose $\boldsymbol {\omega }$ is an extremal vertex in $G(\boldsymbol {\pi })$ . Let $\boldsymbol {\varpi } = \boldsymbol {\pi }\boldsymbol {\omega }^{-1}$ and assume there exists a nontrivial factorization $\boldsymbol {\varpi }= \boldsymbol {\varpi }^{(1)}\boldsymbol {\varpi }^{(2)}$ such that

$$ \begin{align*} L_q(\boldsymbol{\varpi})\cong L_q(\boldsymbol{\varpi}^{(1)})\otimes L_q(\boldsymbol{\varpi}^{(2)}) \end{align*} $$

and every q-factor of $\boldsymbol {\pi }$ adjacent to $\boldsymbol {\omega }$ in $G(\boldsymbol {\pi })$ lies in $G(\boldsymbol {\varpi }^{(1)})$ . Then,

$$ \begin{align*} L_q(\boldsymbol{\pi})\cong L_q(\boldsymbol{\omega}\boldsymbol{\varpi}^{(1)})\otimes L_q(\boldsymbol{\varpi}^{(2)}). \end{align*} $$

Proof Up to arrow dualization, we can assume $\boldsymbol {\omega }$ is a source and, hence,

$$ \begin{align*} L_q(\boldsymbol{\omega})\otimes L_q(\boldsymbol{\omega}') \quad\text{is highest-}\ell\text{-weight for every other } q\text{-factor }\boldsymbol{\omega}'\text{ of }\boldsymbol{\pi}. \end{align*} $$

By Corollary 4.1.4, it suffices to prove that both

$$ \begin{align*} L_q(\boldsymbol{\omega}\boldsymbol{\varpi}^{(1)})\otimes L_q(\boldsymbol{\varpi}^{(2)}) \qquad\text{and}\qquad L_q(\boldsymbol{\varpi}^{(2)})\otimes L_q(\boldsymbol{\omega}\boldsymbol{\varpi}^{(1)}) \end{align*} $$

are highest- $\ell $ -weight. Consider the tensor product

$$ \begin{align*} W_1=L_q(\boldsymbol{\omega})\otimes L_q(\boldsymbol{\varpi}^{(1)})\otimes L_q(\boldsymbol{\varpi}^{(2)}). \end{align*} $$

We claim that $W_1$ is highest- $\ell $ -weight. Indeed, we are assuming that $L_q(\boldsymbol {\varpi }^{(1)})\otimes L_q(\boldsymbol {\varpi }^{(2)})$ is simple and the hypothesis on $G(\boldsymbol {\varpi }^{(1)})$ together with Corollary 4.1.6 implies $L_q(\boldsymbol {\omega })\otimes L_q(\boldsymbol {\varpi }^{(2)})$ is also simple. Also, since $\boldsymbol {\omega }$ is a source, Corollary 4.1.6 implies $L_q(\boldsymbol {\omega })\otimes L_q(\boldsymbol {\varpi }^{(1)})$ is highest- $\ell $ -weight. Hence, the strong version of Theorem 4.1.5 implies $W_1$ is highest- $\ell $ -weight, as well as its quotient $L_q(\boldsymbol {\omega }\boldsymbol {\varpi }^{(1)})\otimes L_q(\boldsymbol {\varpi }^{(2)})$ . These facts also imply $L_q(\boldsymbol {\varpi }^{(2)})\otimes L_q(\boldsymbol {\omega }) \otimes L_q(\boldsymbol {\varpi }^{(1)})$ is highest- $\ell $ -weight, showing that $L_q(\boldsymbol {\varpi }^{(2)})\otimes L_q(\boldsymbol {\omega }\boldsymbol {\varpi }^{(1)})$ is highest- $\ell $ -weight.

Lemma 4.2.2 Suppose $L_q(\boldsymbol {\pi })$ is prime and that $\boldsymbol {\omega }$ is an extremal vertex in $G(\boldsymbol {\pi })$ . Let $\boldsymbol {\varpi } = \boldsymbol {\pi }\boldsymbol {\omega }^{-1}$ . Then, either $L_q(\boldsymbol {\varpi })$ is prime or there exists a nontrivial factorization $\boldsymbol {\varpi }= \boldsymbol {\varpi }^{(1)}\boldsymbol {\varpi }^{(2)}$ such that

$$ \begin{align*} L_q(\boldsymbol{\varpi})\cong L_q(\boldsymbol{\varpi}^{(1)})\otimes L_q(\boldsymbol{\varpi}^{(2)}) \end{align*} $$

and both $\boldsymbol {\varpi }^{(1)}$ and $\boldsymbol {\varpi }^{(2)}$ contain q-factors of $\boldsymbol {\pi }$ adjacent to $\boldsymbol {\omega }$ in $G(\boldsymbol {\pi })$ .

We can now give the proof mentioned in Remark 3.5.2. Write $G=G(\boldsymbol {\pi })$ , let $\boldsymbol {\omega }=v$ , and $\boldsymbol {\varpi }=\boldsymbol {\pi }\boldsymbol {\omega }^{-1}$ . In particular, $G_{\mathcal V\setminus \{v\}}=G(\boldsymbol {\varpi })$ . Since G is connected by Proposition 3.4.1, v must be monovalent and, hence, there exists a unique q-factor $\boldsymbol {\omega }'$ of $\boldsymbol {\varpi }$ such that $L_q(\boldsymbol {\omega })\otimes L_q(\boldsymbol {\omega }')$ is reducible. In particular, $\boldsymbol {\omega }$ is extremal in G. The claim then follows immediately from Lemma 4.2.2.

4.3 A key highest- $\ell $ -weight criterion and Theorem 3.5.4

We now establish a criterion for a tensor product to be highest- $\ell $ -weight which is the heart of the proof of Theorem 3.5.4 and will also be used to deduce further criteria which will be used in the proof of Theorem 3.5.5.

Proof We write the details only in case (i) holds since the other case is similar. So, assume that V is not highest- $\ell $ -weight. In particular, there exists $\boldsymbol {\xi }\in \mathcal {P}^+$ such that $\boldsymbol {\xi }<\boldsymbol {\lambda }\boldsymbol {\nu }$ together with an epimorphism $V\xrightarrow {f} L_q(\boldsymbol {\xi })$ . Therefore, there also exists an epimorphism

$$ \begin{align*} L_q(\boldsymbol{\mu})\otimes V=L_q(\boldsymbol{\mu})\otimes L_q(\boldsymbol{\lambda})\otimes L_q(\boldsymbol{\nu})\xrightarrow{\operatorname{id}_{L_q(\boldsymbol{\mu})}\otimes f}L_q(\boldsymbol{\mu})\otimes L_q(\boldsymbol{\xi}). \end{align*} $$

On the other hand, since $L_q(\boldsymbol {\lambda })\otimes L_q(\boldsymbol {\mu })$ is highest- $\ell $ -weight, there exist monomorphisms

$$ \begin{align*} L_q(\boldsymbol{\lambda}\boldsymbol{\mu})\xrightarrow{g} L_q(\boldsymbol{\mu})\otimes L_q(\boldsymbol{\lambda}) \ \ \text{and}\ \ L_q(\boldsymbol{\lambda}\boldsymbol{\mu})\otimes L_q(\boldsymbol{\nu})\xrightarrow{g\otimes\operatorname{id}_{L_q(\boldsymbol{\nu})}} L_q(\boldsymbol{\mu})\otimes L_q(\boldsymbol{\lambda})\otimes L_q(\boldsymbol{\nu}). \end{align*} $$

Lemma 2.6.2 implies the composition

$$ \begin{align*} L_q(\boldsymbol{\lambda}\boldsymbol{\mu})\otimes L_q(\boldsymbol{\nu})\xrightarrow{g\otimes\operatorname{id}_{L_q(\boldsymbol{\nu})}}L_q(\boldsymbol{\mu})\otimes L_q(\boldsymbol{\lambda})\otimes L_q(\boldsymbol{\nu})\xrightarrow{\operatorname{id}_{L_q(\boldsymbol{\mu})}\otimes f}L_q(\boldsymbol{\mu})\otimes L_q(\boldsymbol{\xi}) \end{align*} $$

is nonzero. Then, since $L_q(\boldsymbol {\lambda }\boldsymbol {\mu })\otimes L_q(\boldsymbol {\nu })$ is highest- $\ell $ -weight, the image of its highest- $\ell $ -weight vector under this composition must be a nonzero vector with $\ell $ -weight $\boldsymbol {\lambda }\boldsymbol {\mu }\boldsymbol {\nu }$ . However, since $\boldsymbol {\xi }<\boldsymbol {\lambda }\boldsymbol {\nu }$ , we also have $\boldsymbol {\xi }\boldsymbol {\mu }<\boldsymbol {\lambda }\boldsymbol {\mu }\boldsymbol {\nu }$ and, hence, there is no vector in $L_q(\boldsymbol {\mu })\otimes L_q(\boldsymbol {\xi })$ with $\ell $ -weight $\boldsymbol {\lambda }\boldsymbol {\mu }\boldsymbol {\nu }$ , yielding a contradiction.

One of the applications of the above result gives a partial answer to the following question. Suppose $\boldsymbol {\pi },\boldsymbol {\pi }',\boldsymbol {\varpi }\in \mathcal P^+$ are such that $L_q(\boldsymbol {\pi })\otimes L_q(\boldsymbol {\pi }')$ is highest- $\ell $ -weight and $\boldsymbol {\varpi }$ divides $\boldsymbol {\pi }$ . Under which further assumptions $L_q(\boldsymbol {\pi }\boldsymbol {\varpi }^{-1})\otimes L_q(\boldsymbol {\pi }')$ is also highest- $\ell $ -weight? A similar question can be made in the case $\boldsymbol {\varpi }$ divides $\boldsymbol {\pi }'$ .

Corollary 4.3.3 Let $\boldsymbol {\pi }',\boldsymbol {\pi }"\in \mathcal {P}^+$ with dissociate factorizations and $\boldsymbol {\pi }=\boldsymbol {\pi }'\boldsymbol {\pi }"$ . Let also $G=G(\boldsymbol {\pi }), G'=G(\boldsymbol {\pi }'), G"=G(\boldsymbol {\pi }")$ , and suppose $\boldsymbol {\omega }',\boldsymbol {\omega }"\in \mathcal {P}^+$ satisfy

$$ \begin{align*} \boldsymbol{\omega}' \text{ is a source in } G', \quad \boldsymbol{\omega}" \text{ is a sink in } G", \quad\text{and}\quad (\boldsymbol{\omega}",\boldsymbol{\omega}') \in \mathcal{A}_G. \end{align*} $$

Then, $L_q(\boldsymbol {\pi }')\otimes L_q(\boldsymbol {\pi }")$ is not highest- $\ell $ -weight.

Proof The first two assumptions, together with Corollary 4.1.6, imply that

$$ \begin{align*} L_q(\boldsymbol{\omega}')\otimes L_q(\boldsymbol{\pi}'(\boldsymbol{\omega}')^{-1})\quad\textrm{and}\quad L_q(\boldsymbol{\pi}"(\boldsymbol{\omega}")^{-1})\otimes L_q(\boldsymbol{\omega}") \quad\text{are highest-}\ell\text{-weight.} \end{align*} $$

On the other hand, the last assumption implies that $L_q(\boldsymbol {\omega }')\otimes L_q(\boldsymbol {\omega }")$ is not highest- $\ell $ -weight. Letting $\boldsymbol {\lambda }=\boldsymbol {\omega }',\boldsymbol {\mu }=\boldsymbol {\pi }'(\boldsymbol {\omega }')^{-1}$ and $\boldsymbol {\nu }=\boldsymbol {\omega }"$ , Proposition 4.3.1(i) implies $L_q(\boldsymbol {\pi }')\otimes L_q(\boldsymbol {\omega }")$ is not highest- $\ell $ -weight. Then, an application of Proposition 4.3.1(ii) with $\boldsymbol {\mu } = \boldsymbol {\pi }"(\boldsymbol {\omega }")^{-1}, \boldsymbol {\nu } = \boldsymbol {\omega }"$ , and $\boldsymbol {\lambda }=\boldsymbol {\pi }'$ completes the proof.

Theorem 3.5.7 is easily deduced from Corollary 4.3.3. Moreover, we can also give the following proof.

4.4 Highest- $\ell $ -weight criteria via monotonic paths

Recall (2.2.6).

Lemma 4.4.1 Let $\boldsymbol {\pi }\in \mathcal {P}^+$ and suppose G is a pseudo q-factorization graph over $\boldsymbol {\pi }$ . Given $\boldsymbol {\varpi }\in \mathcal {V}_G$ , consider

$$ \begin{align*} \boldsymbol{\pi}_\pm=\prod\limits_{\boldsymbol{\omega}\in\mathcal{N}_G^\pm(\boldsymbol{\varpi})}\boldsymbol{\omega}'. \end{align*} $$

Then, the following tensor products are highest- $\ell $ -weight:

$$ \begin{align*} L_q(\boldsymbol{\pi}_+)\otimes L_q(\boldsymbol{\pi}\boldsymbol{\pi}_+^{-1})\quad\textrm{and}\quad L_q(\boldsymbol{\pi}\boldsymbol{\pi}_-^{-1})\otimes L_q(\boldsymbol{\pi}_-). \end{align*} $$

The next lemma will play a role in the proofs of Proposition 4.5.3 and Theorem 3.5.5.

Lemma 4.4.2 Let $\boldsymbol {\pi },\boldsymbol {\pi '}\in \mathcal {P}^+$ and suppose G and $G'$ are pseudo q-factorization graphs over $\boldsymbol {\pi }$ and $\boldsymbol {\pi '}$ , respectively. Let $\boldsymbol {\varpi }\in \mathcal {V}_G$ and $\boldsymbol {\varpi }'\in \mathcal {V}_{G'}$ and consider

$$ \begin{align*} \boldsymbol{\pi}_+=\prod\limits_{\boldsymbol{\omega}\in\mathcal{N}^+_G(\boldsymbol{\varpi})}\boldsymbol{\omega}\quad\textrm{and}\quad\boldsymbol{\pi}^{\prime}_-=\prod\limits_{\boldsymbol{\omega}\in\mathcal{N}^-_{G'}(\boldsymbol{\varpi}')}\boldsymbol{\omega}. \end{align*} $$

If $V=L_q(\boldsymbol {\pi })\otimes L_q(\boldsymbol {\pi }')$ is highest- $\ell $ -weight, so are the following tensor products:

$$ \begin{align*} L_q(\boldsymbol{\pi}_+)\otimes L_q(\boldsymbol{\pi}'),\quad L_q(\boldsymbol{\pi})\otimes L_q(\boldsymbol{\pi}^{\prime}_-),\quad\textrm{and}\quad L_q(\boldsymbol{\pi}_+)\otimes L_q(\boldsymbol{\pi}^{\prime}_-). \end{align*} $$

4.5 A highest- $\ell $ -weight criterion from duality

We now deduce the main technical part behind the proof of Proposition 3.5.9 which will also be used for proving Theorem 3.5.5.

Proposition 4.5.1 Let $\boldsymbol {\lambda },\boldsymbol {\mu },\boldsymbol {\nu }\in \mathcal {P}^+$ . Let also $V = L_q(\boldsymbol {\lambda })\otimes L_q(\boldsymbol {\nu })^*$ ,

$$ \begin{align*} T_1=L_q(\boldsymbol{\lambda}\boldsymbol{\mu})\otimes L_q(\boldsymbol{\nu}),\quad U_1=L_q(\boldsymbol{\lambda})\otimes L_q(\boldsymbol{\mu}), \quad W_1=L_q(\boldsymbol{\mu})\otimes L_q(\boldsymbol{\nu}), \end{align*} $$

$$ \begin{align*} T_2=L_q(\boldsymbol{\lambda})\otimes L_q(\boldsymbol{\mu}\boldsymbol{\nu}),\quad U_2=L_q(\boldsymbol{\mu})\otimes L_q(\boldsymbol{\nu}),\quad W_2=L_q(\boldsymbol{\lambda})\otimes L_q(\boldsymbol{\mu}). \end{align*} $$

Then, $W_i$ is highest- $\ell $ -weight provided $T_i$ and $U_i$ are highest- $\ell $ -weight, $i\in \{1,2\}$ , and V is simple.