2.1 Totally nonnegative Grassmannians

The tnn Grassmannian $\mathrm {Gr}(k,n)_{\ge 0}$ is the subset of the real Grassmann variety $\mathrm {Gr}(k,n)$ represented by the subspaces whose Plücker coordinates have all the same sign. Postnikov [Reference PostnikovPos06] defined a cellular decomposition of $\mathrm {Gr}(k,n)_{\ge 0}$ as follows. For $L\in \mathrm {Gr}(k,n)$ and a k-element subset $I\subset [n]=\{1,\dots ,n\}$ , let $X_I(L)$ be the Ith Plücker coordinate of L. We define

$$\begin{align*}{\mathcal M}(L)=\left\{I\subset\binom{[n]}{k}:\ X_I(L)\ne 0\right\}. \end{align*}$$

Then ${\mathcal M}(L)$ is a matroid attached to L. If $L\in \mathrm {Gr}(k,n)_{\ge 0}$ , then the matroid ${\mathcal M}(L)$ is called a positroid.

The importance of this notion is explained by the following theorem due to Postnikov [Reference PostnikovPos06].

Hence, one gets a cellular decomposition of the tnn Grassmannian labeled by the positroids. It is thus natural to ask how to label the positroids ${\mathcal P}(k,n)$ and how to compute the dimension of the cell $\Pi ({\mathcal M})$ for ${\mathcal M}\in {\mathcal P}(k,n)$ .

2.2 Grassmann necklaces

There are several ways to parametrize the elements of ${\mathcal P}(k,n)$ . The one providing the bridge between the theory of tnn Grassmannians and quiver Grassmannians for (equioriented) cyclic quivers is the following one (see [Reference PostnikovPos06]).

We note that for $a=n$ the last condition is understood as $I_n\subset I_1\cup \{n\}$ . In other words, the condition on the sets $I_a$ can be written as $i\in I_a\setminus \{a\}$ implies $i\in I_{a+1}$ (which works for $a=n$ as well). There is a slightly different version of Definition 2.4 (see, e.g., [Reference Knutson, Lam and SpeyerKLS13]). Namely, given a collection $(I_a)_{a\in [n]}\in \mathcal {GN}_{k,n}$ , we define $J_a=\{i-a:\ i\in I_a\}$ , where $i-a$ is understood as an element of $[n]$ , which is equal to $a-n$ modulo n. The resulting collections $(J_a)_{a\in [n]}$ are called juggling patterns in [Reference Knutson, Lam and SpeyerKLS13] (modulo the overall change $j\to n+1-j$ of the elements of $J_a$ ). Clearly, one can put forward the following definition.

The set of $(k,n)$ Grassmann necklaces can be equipped with a partial order. For two elements $I,J\in \binom {[n]}{k}$ such that $I=(i_1<\cdots <i_k)$ , $J=(j_1<\cdots <j_k)$ , we write $I\le J$ if $i_u\le j_u$ for all $u \in [k]$ . Now, for a number $a\in [n]$ , we consider the rotated order

(2.1) $$ \begin{align} a<_a a+1 <_a \cdots <_a n <_a 1 <_a \cdots <_a a-1 \end{align} $$

on the set $[n]$ . This order induces the order $<_a$ on the set $\binom {[n]}{k}$ . Now, for two $(k,n)$ Grassmann necklaces ${\mathcal I}=(I_1,\dots ,I_n)$ and ${\mathcal J}=(J_1,\dots ,J_n)$ , we write ${\mathcal I}\le {\mathcal J}$ if $I_a\le _a J_a$ for all $a\in [n]$ .

Example 2.7 Let $k=1$ and $n=3$ . Given a Grassmann necklace $(I_1=\{i_1\},I_2=\{i_2\},I_3=\{i_3\})\in \mathcal {GN}_{1,3}$ , we represent such an element by $i_1i_2i_3$ . In this case, the Hasse diagram of the poset $(\mathcal {GN}_{1,3},\leq )$ is the following:

Given a positroid ${\mathcal M}\in {\mathcal P}(k,n)$ , we define the corresponding Grassmann necklace ${\mathcal I}({\mathcal M})$ by the formula

(2.2) $$ \begin{align} {\mathcal I}({\mathcal M})_a=\min{_a} \{J\in{\mathcal M}\}, \end{align} $$

where $\min _a$ is the minimum with respect to the order $\le _a$ .

2.3 Bounded affine permutations

We recall here briefly the definition of bounded affine permutations and their relation with Grassmann necklaces. More details can be found in [Reference Knutson, Lam and SpeyerKLS13, Section 3] and [Reference LamLam16, Section 6].

Recall that each stratum $\Pi ({\mathcal M})$ is a cell ${\mathbb R}_{> 0}^d$ . In order to give a formula for the dimension d of the cell, we use one more parametrization via the bounded affine permutations. A $(k,n)$ affine permutation (not yet bounded) is a bijection $f:{\mathbb Z}\to {\mathbb Z}$ satisfying the following properties:

• • $f(i+n)=f(i)+n$ for all $i\in {\mathbb Z}$ ,

• • $\sum _{i=1}^n (f(i)-i)=kn$ .

In particular, there is a distinguished $(k,n)$ affine permutation $\mathrm {id}_k$ given by $\mathrm {id}_k(i)=i+k$ . The length of an affine permutation is defined as

$$\begin{align*}l(f)=|\{(i,j)\in[n]\times{\mathbb Z}:\ i<j \text{ and } f(i)>f(j)\}|. \end{align*}$$

We note that the set of $(0,n)$ affine permutations is a group isomorphic to the affine Weyl group $W_n$ of type $A_{n-1}^{(1)}$ . For general $k=1,\dots ,n$ , the group $W_n$ acts freely and transitively on the set of $(k,n)$ affine permutations; the action of the permutation $s_i=(i,i+1)\in W_n$ , for $i=0,\dots ,n-1$ , permutes the values $f(i+rn)$ and ${f(i+rn+1)}$ for all $r\in {\mathbb Z}$ . This allows to identify the set of $(k,n)$ affine permutations with $W_n$ by sending $w\in W_n$ to $w.\mathrm {id}_k$ . We thus obtain an order $\le $ on the set of $(k,n)$ affine permutations coming from the Bruhat order on $W_n$ . For example, the unique minimal element is $\mathrm {id}_k$ .

A $(k,n)$ bounded affine permutation is a $(k,n)$ affine permutation subject to the extra condition

$$\begin{align*}i\le f(i)\le i+n \text{ for all } i\in{\mathbb Z}. \end{align*}$$

We denote the set of $(k,n)$ bounded affine permutations by ${\mathcal B}_{k,n}$ .

It is shown in [Reference Knutson, Lam and SpeyerKLS13] that ${\mathcal B}_{k,n}$ is a lower-order ideal in the set of $(k,n)$ affine permutations (unbounded). For $f,g\in {\mathcal B}_{k,n}$ , we write $f\le g$ for the induced order. The Grassmann necklace ${\mathcal I}(f)=(I_1,\dots ,I_n)$ for $f\in {\mathcal B}_{k,n}$ is defined by the formula

$$\begin{align*}I_a=\{f(b):\ b<a \text{ and } f(b)\ge a\} \quad\mod n. \end{align*}$$

Example 2.9 The Grassmann necklace corresponding to $\mathrm {id}_k$ is the one defined by

$$\begin{align*}I_a=(a, a+1, \ldots, a+k-1) \qquad (a\in[n]). \end{align*}$$

By [Reference LamLam16, Theorem 6.2], this defines an order-preserving bijection between the set ${\mathcal B}_{k,n}$ and the set of $(k,n)$ Grassmann necklaces.

In the opposite direction, given a Grassmann necklace ${\mathcal I}$ , we define the bounded affine permutation $f=f({\mathcal I})$ as follows: if $a\notin I_a$ , then $f(a)=a$ . If $a\in I_a$ , then $I_{a+1}=I_a\setminus \{a\}\cup \{b\}$ . We define $f(a)=c$ , where $b\equiv c \quad\mod n$ and $a<c\le a+n$ .

3.1 Quivers and representations

A finite quiver Q consists of a finite set of vertices $Q_0$ and a finite set of arrows $Q_1$ . Each $\alpha \in Q_1$ has a unique source and target $i,j \in Q_0$ , and we write $(\alpha : i \to j)$ . A finite-dimensional Q representation M is a pair of tuples $(M^{(i)})_{i \in Q_0}$ and $(M_\alpha )_{\alpha \in Q_1}$ , where each $M^{(i)}$ is a finite-dimensional $\mathbb {C}$ -vector space and each $M_\alpha $ is a linear map from $M^{(i)}$ to $M^{(j)}$ . The notion of a subrepresentation will be fundamental to us: a tuple of finite-dimensional ${\mathbb C}$ -vector spaces $N=(N^{(i)})_{i \in Q_0}$ is a subrepresentation of $M=((M^{(i)})_{i\in Q_0},(M_\alpha )_{\alpha \in Q_1})$ if $N^{(i)}\subset M^{(i)}$ for all $i\in Q_0$ and $M_\alpha N^{(i)} \subset N^{(j)}$ for all $\alpha :i\to j \in Q_1$ .

A morphism from the Q representation M to the Q representation N is a tuple of linear maps $\varphi = (\varphi _i)_{i\in Q_0} \in \prod _{i \in Q_0} \mathrm {Hom}_{\mathbb {C}}(M^{(i)},N^{(i)})$ such that:

By $\mathrm {Hom}_Q(M,N)$ , we denote the set of all Q morphisms from M to N. The category of finite-dimensional Q representations over $\mathbb {C}$ is denoted by $\mathrm {rep}_{\mathbb {C}}(Q)$ . The dimension vector of $M \in \mathrm {rep}_{\mathbb {C}}(Q)$ is

$$\begin{align*}\mathbf{d} := \big(\! \dim_{\mathbb{C}} M^{(i)} \big)_{i \in Q_0} \in \mathbb{Z}_{\geq 0}^{Q_0}. \end{align*}$$

3.2 The path algebra of a quiver

A path p in a quiver Q is a concatenation of consecutive arrows. We define the source of a path as the source of its first arrow, and its target is the target of the last arrow. The path algebra $\mathbb {C}Q$ has all paths in Q as basis, and the multiplication $*$ of two paths p and $p'$ is defined via concatenation: if the target of p is the source of $p'$ , then $p'*p:= p' \circ p$ . Otherwise, the product is zero.

Using paths in Q, we can define a set of relations R on the objects of $\mathrm {rep}_{\mathbb {C}}(Q)$ . Let $I \subset \mathbb {C}Q$ be the ideal generated by the relations in R. Then there is an equivalence of categories between $\mathrm {rep}_{\mathbb {C}}(Q,I)$ , i.e., representations satisfying the relations in I (called bounded quiver representations) and $\mathrm {mod}_{\mathbb {C}}(\mathbb {C}Q/I)$ , i.e., modules over the so-called bounded path algebra [Reference SchifflerSch14, Theorem 5.4].

The injective bounded representation $I_k$ at the vertex $k \in Q_0$ consists of the vector spaces $V^{(j)}$ for $j \in Q_0$ with a basis indexed by equivalence classes of paths (in $\mathbb {C}Q/I$ ) from j to k. The linear map $V_\alpha $ along the arrow $(\alpha : i \to j) \in Q_1$ sends a basis element of $V^{(i)}$ indexed by the equivalence class of a path p to a basis element of $V^{(j)}$ which is indexed by another equivalence class with representative $p'$ such that $p = \alpha \circ p'$ . A basis element of $V^{(i)}$ is sent to zero by the map along the arrow $(\alpha : i \to j)$ if the paths in the equivalence class indexing this basis element do not factor through $\alpha $ .

3.3 The equioriented Cycle

Let $\Delta _n$ denote the equioriented cycle on n vertices, where the orientation is chosen in such a way that $1\to 2$ is an arrow. Then the set of vertices and the set of arrows are both in bijection with $\mathbb {Z}_n := \mathbb {Z}/n \mathbb {Z}$ . Unless specified differently, we consider all indices of vertices and arrows modulo n. Given a representation $M\in \mathrm {rep}_{\mathbb {C}}(\Delta _n)$ , we write $M_{\alpha _i}$ for $M_{i\to i+1}$ for any $i\in {\mathbb Z}_n$ . We call $M \in \mathrm {rep}_{\mathbb {C}}(\Delta _n)$ nilpotent if all concatenations of the maps of M along cyclic paths vanish beyond a certain length of the paths.

Every point M of the affine variety

$$\begin{align*}\mathrm{R}_{\mathbf{n}}(\Delta_n):= \bigoplus_{i \in \mathbb{Z}_n} \mathrm{Hom}_{\mathbb{C}}(\mathbb{C}^n,\mathbb{C}^n)\end{align*}$$

parametrizes a $\Delta _n$ representation of dimension vector $\mathbf {n}:= (n,\dots ,n) \in \mathbb {Z}^n$ . With a fixed ordered basis $(e_j)_{j\in [n]}$ of $\mathbb {C}^n$ , each $g \in \mathrm {G}_{\mathbf {n}} := \prod _{i \in \mathbb {Z}_n} GL_n(\mathbb {C})$ acts on $\mathrm { R}_{\mathbf {n}}(\Delta _n)$ via conjugation

$$\begin{align*}g.M := \big( g_{i+1} M_{\alpha_i} g_i^{-1} \big)_{i \in {\mathbb Z}_n}.\end{align*}$$

The automorphism group $\mathrm {Aut}_{\Delta _n}(M)$ of a $\Delta _n$ representation $M \in \mathrm {R}_{\mathbf {n}}(\Delta _n)$ is its stabilizer in $\mathrm {G}_{\mathbf {n}}$ .

Let M be a Q representation. A basis of M is a basis B of the underlying vector space $\bigoplus _{i\in \ Q_0}M^{(i)}$ . In the case of $Q=\Delta _n$ , we will always pick bases B of M compatible with the ${\mathbb Z}_n$ -grading on $\bigoplus _{i\in {\mathbb Z}_n}M^{(i)}$ : $B=\bigcup _{i\in {\mathbb Z}_n} B^{(i)}$ , where $B^{(i)}=\{w_1^{(i)}, \ldots , w_{d_i}^{(i)}\}$ is a basis for $M^{(i)}$ .

Example 3.3 Let $U(i;n)$ be the $\Delta _n$ representation given by

$$\begin{align*}U(i;n)^{(j)}={\mathbb C}, \qquad U(i;n)_{\alpha_j}=\left\{ \begin{array}{ll} \mathrm{id}_{{\mathbb C}} , & \mbox{ if }j\neq i,\\ 0, & \mbox{ if }j=i. \end{array} \right. \end{align*}$$

If we denote by $w^{(j)}$ a generator of $U(i;n)^{(j)}$ (that is, any nonzero element), then the corresponding coefficient quiver is just an equioriented type $A_n$ Dynkin quiver:

$$\begin{align*}w^{(i+1)}\to w^{(i+2)}\to\cdots\to w^{(i-1)}\to w^{(i)}. \end{align*}$$

In [Reference Lanini and PützLP20], quiver Grassmannians for nilpotent $\Delta _n$ -representations are investigated. More precisely, they are equipped with torus actions, and these actions exploited to obtain cellular decompositions and study equivariant cohomology.

Given a fixed basis B of $M\in \mathrm {rep}_{\mathbb {C}}(\Delta _n)$ , a grading of M is simply a map $\mathrm {wt} : B \to \mathbb {Z}^B$ . This induces a $\mathbb {C}^*$ action on the vector spaces of M, defined on the basis B as follows and then extended by linearity:

(3.1) $$ \begin{align} \lambda.b := \lambda^{\mathrm{wt}(b)}\cdot b\qquad (b\in B, \lambda\in{\mathbb C}^*). \end{align} $$

With some additional assumptions about the grading (see [Reference Lanini and PützLP20, Section 5.1]), the $\mathbb {C}^*$ action extends to the quiver Grassmannian $\mathrm {Gr}_{\mathbf {e}}(M)$ with finitely many fixed points

$$\begin{align*}\{ L_1, \dots, L_m \} =: \mathrm{Gr}_{\mathbf{e}}(M)^{\mathbb{C}^*}\end{align*}$$

indexed by appropriate subquivers of $Q(M,B)$ (see [Reference Cerulli IrelliCI11, Proposition 1]). Moreover, the ${\mathbb C}^*$ action induces an $\alpha $ -partition of $\mathrm {Gr}_{\mathbf {e}}(M)$ into the attracting sets of the fixed points

$$\begin{align*}W_L := \big\{ U \in \mathrm{Gr}_{\mathbf{e}}(M) : \lim_{\lambda \to 0} \lambda.U = L \big\}, \end{align*}$$

i.e., there exists a total order on the fixed point set such that $\bigsqcup _{i=1}^s W_{L_i} $ is closed in $\mathrm {Gr}_{\mathbf {e}}(M)$ for all $s \in [m]$ . All the $W_L$ ’s are isomorphic to affine spaces by [Reference Lanini and PützLP20, Theorem 5.6].

4.1 The automorphism group of $\boldsymbol{U_{[n]}}$

The explicit realization of the group $\mathrm {Aut}_{\Delta _n}(U_{[n]})$ as a subgroup of $\mathrm { G}_{\mathbf {n}}$ depends on the basis of $U_{[n]}$ .

There are two special ordered bases which we use throughout this paper.

Observe that the choice of a basis corresponds to a certain realization of $U_{[n]}$ as a point in $\mathrm {R}_{\mathbf {n}}(\Delta _n)$ . If it is clear from the context that we use our preferred ordered basis, we just refer to it as basis from now on.

Remark 4.4 The second realization of $U_{[n]}$ from the definition above leads to the following realization of $X(k,n)$ (juggling patterns style). Let $W'$ be an n-dimensional ${\mathbb C}$ -vector space, and let $(v_1, \ldots , v_n)$ be a basis of $W'$ , then

$$\begin{align*}X(k,n)= \left\{ (V_i)\in \prod_{i\in {\mathbb Z}_n}\mathrm{Gr}_k(W')\mid s_1(V_i)\subseteq V_{i+1} \right\},\end{align*}$$

where $s_1(v_j)=v_{j+1}$ for any $j\neq n$ and $s_1(v_n)=0$ .

If $M \in \mathrm {R}_{\mathbf {n}}(\Delta _n)$ , then its endomorphism algebra $\mathrm {End}_{\Delta _n}(M)$ is defined as the set of matrix tuples $ E = (E_i)_{i \in \mathbb {Z}_n} \prod _{i \in \mathbb {Z}_n} \mathrm {M}_n(\mathbb {C})$ such that

$$\begin{align*}E_{i+1}M_i = M_i E_i \quad \mathrm{for} \ \mathrm{all} \ i \in \mathbb{Z}_n. \end{align*}$$

Proposition 4.5 With respect to the basis $\bigcup _{i\in {\mathbb Z}_n}\{v_1^{(i)}, \ldots , v_n^{(i)}\}$ , the elements of the endomorphism algebra $\mathrm { End}_{\Delta _n}(U_{[n]})$ are exactly the matrix tuples $ E = (E_i)_{i \in \mathbb {Z}_n}$ with

$$\begin{align*}E_i = \begin{pmatrix} e^{(i)}_{1,1} & & & &\\ e^{(i)}_{2,1}& e^{(i-1)}_{1,1} & & &\\ \vdots & \vdots & \ddots& &\\ e^{(i)}_{n-1,1} & e^{(i-1)}_{n-2,1} & \ldots & e^{(i-n+2)}_{1,1} &\\ e^{(i)}_{n,1} & e^{(i-1)}_{n-1,1} & \ldots & e^{(i-n-2)}_{2,1} & e^{(i-n+1)}_{1,1} \end{pmatrix}, \end{align*}$$

where $e^{(i)}_{k,1} \in \mathbb {C}$ for all $i \in \mathbb {Z}_n$ , $k \in [n]$ . In particular, $\dim _{\mathbb C} \mathrm { End}_{\Delta _n}(U_{[n]})=n^2$ .

Proof By definition of $\mathrm {End}_{\Delta _n}(U_{[n]})$ , we have that $(E_i)_{i\in {\mathbb Z}_n}$ if and only if

$$\begin{align*}E_{i+1} s_1 = s_1 E_i \quad \mathrm{for} \ \mathrm{all} \ i \in \mathbb{Z}_n, \end{align*}$$

that is,

(4.1) $$ \begin{align} E_{i+1} s_1(v_l^{(i)}) = s_1 E_i(v_l^{(i)}) \quad \mathrm{for} \ \mathrm{all} \ i,l \in \mathbb{Z}_n. \end{align} $$

Let us we write $e^{(i)}_{k,l}:= (E_i)_{k,l}$ , so that $E_i(v_l^{(i)})=\sum _{k=1}^n e_{k,l}^{(i)}v_k^{(i)}$ . It is then easy to see that equations (4.1) are equivalent to

$$\begin{align*}e^{(i)}_{k,l} = e^{(i+1)}_{k+1,l+1},\quad e^{(i)}_{k,n}= 0, \quad e^{(i)}_{n,l}= 0, \qquad \mathrm{for} \ \mathrm{all } \ k,l \in [n-1]. \end{align*}$$

From the previous equations, it follows by induction on $n-l$ that $e_{k,l}^{(i)}=0$ for any $l>k$ , and by induction on l that $e_{k,l}^{(i)}=e_{k+1,l+1}^{(i+1)}$ . This implies that the $E_i$ ’s are of the desired form.▪

4.2 Geometric properties of the main object

We prove here, by applying quiver representation theory results, geometric properties of $X(k,n)$ .

Proof By [Reference PützPue20, Lemma 4.10], the irreducible components of $X(k,n)$ are parametrized by the set

$$\begin{align*}\left\{ p = (p_i)_{i \in \mathbb{Z}_n} \in \prod_{i \in \mathbb{Z}_n} \{ 0,1 \} \ : \ \sum_{i \in \mathbb{Z}_n} p_i = k \right\}, \end{align*}$$

and they are all of dimension $k(n-k)$ . The above set is in bijection with $\binom {[n]}{k}$ , i.e., the set containing all k-element subsets of $[n] := \{1,\dots ,n\}$ .

Example 4.9 Let $k=1$ and $n=3$ . Observe that, in this case,

$$\begin{align*}X(1,3)\simeq X:=\left\{ \left( \begin{bmatrix} a_i\\b_i\\c_i \end{bmatrix} \right)\in\prod_{i\in{\mathbb Z}_3}\mathbb{P}^2\mid \mathrm{pr}_i\left(\begin{matrix} a_i\\b_i\\c_i \end{matrix}\right)\in{\mathbb C}\left(\begin{matrix} a_{i+1}\\b_{i+1}\\c_{i+1} \end{matrix}\right), \ i\in{\mathbb Z}_3 \right\}. \end{align*}$$

The three irreducible components are

$$\begin{align*}X_i=\left\{ \left( \begin{bmatrix} a_i\\b_i\\c_i \end{bmatrix} \right)\in X\mid \begin{bmatrix} a_i\\b_i\\c_i \end{bmatrix}=[b^{(i)}_i] \right\}\qquad (i\in{\mathbb Z}_3), \end{align*}$$

where $[b^{(i)}_i]$ , according to Definition 4.3(1), denotes the class of the ith standard basis vector of ${\mathbb C}^3$ . For example,

$$ \begin{align*} X_1&= \left\{ \left( \begin{bmatrix} 1\\0\\0 \end{bmatrix},\begin{bmatrix} a_2\\b_2\\c_2 \end{bmatrix},\begin{bmatrix} a_3\\0\\c_3 \end{bmatrix} \right)\in\prod_{i\in{\mathbb Z}_3}\mathbb{P}^2\mid a_2c_3-a_3c_2=0 \right\}. \end{align*} $$

We see immediately that $X_1$ is a projective variety of dimension 2.

Proof Let $B = \cup _{i \in \mathbb {Z}_n} B^{(i)}$ be the first basis of $U_{[n]}$ from Definition 4.3. We fix the weight function $\mathrm {wt}(b^{(i)}_j):= j$ for all $i \in \mathbb {Z}_n$ and all $j \in [n]$ . Hence, by [Reference Cerulli IrelliCI11, Proposition 1], the $\mathbb {C}^*$ fixed points are parametrized by the elements of

$$\begin{align*}\left\{ I = (I_j)_{j \in \mathbb{Z}_n} \in \prod_{i \in \mathbb{Z}_n} \binom{[n]}{k} \ : \ I_j \setminus\{j\} \subset I_{j+1} \ \mathrm{for} \ \mathrm{all} \ j \in \mathbb{Z}_n \right\}. \end{align*}$$

This set coincides with the set $\mathcal {GN}_{k,n}$ of Grassmannian necklaces. For a Grassmann necklace ${\mathcal I}=(I_1,\dots ,I_n)$ , the ath component $p({\mathcal I})_a$ of the corresponding $\mathbb {C}^*$ fixed point $p({\mathcal I})$ is a linear span of the basis vectors $b^{(a)}_i$ , $i\in I_a$ .

With this $\mathbb {C}^*$ action, not all attracting sets of the fixed points are isomorphic to affine spaces. For this reason, we switch to the second basis of $U_{[n]}$ from Definition 4.3, but keep the same weight function. Observe that now the weight difference along each arrow of the coefficient quiver is one, whereas it was zero for the first choice of a basis of $U_{[n]}$ . Now, by [Reference Cerulli IrelliCI11, Proposition 1], the fixed points of the induced $\mathbb {C}^*$ action are exactly the juggling patterns as in Definition 2.5. Using the second basis of $U_{[n]}$ and the $\mathbb {C}^*$ action described above, the first part is a special case of [Reference PützPue20, Theorem 4.13].▪

For a point $U \in \mathrm {Gr}_{\mathbf {k}}(M)$ , the isomorphism class $\mathcal {S}_U$ in the quiver Grassmannian is called stratum and is irreducible by [Reference Cerulli Irelli, Feigin and ReinekeCFR12, Lemma 2.4].

Remark 4.12 The closures of the top-dimensional strata are the irreducible components of $X(k,n)=\mathrm {Gr}_{\mathbf {k}}(U_{[n]})$ , and by [Reference PützPue20, Lemma 4.10], these strata have the representatives

$$\begin{align*}U_J = \bigoplus_{j \in J} U(j;n) \quad \mathrm{for} \ J \in \binom{[n]}{k}. \end{align*}$$

5.1 The largest torus acting faithfully

The largest torus acting faithfully on the vector spaces of $U_{[n]}=\bigoplus _{i \in \mathbb {Z}_n} U(i;n)$ is $n^2$ -dimensional, since each vector space of $U_{[n]}$ over $i \in \mathbb {Z}_n$ is n-dimensional and we can act on each basis vector with a different parameter. However, the maximal torus whose action extends to $X(k,n)$ is much smaller:

Proof Assume that we have a weight tuple such that the corresponding T action extends to each quiver Grassmannian associated with $U_{[n]}$ . Then this weight tuple has a fixed weight difference along each arrow of $\Delta _n$ , i.e., if there are two arrows $b_1 \to b_1'$ and $b_2 \to b_2'$ in $Q(U_{[n]},B)$ with the same underlying arrow of $\Delta _n$ , then

$$\begin{align*}\mathbf{wt}(b_1') - \mathbf{wt}(b_1) = \mathbf{wt}(b_2') - \mathbf{wt}(b_2)\end{align*}$$

(see the proof of [Reference Lanini and PützLP20, Lemma 5.10]). The above property of the weight tuples is equivalent to the condition that the grading is constructible from the weights of the predecessor free points in $Q(U_{[n]},B)$ and the weights of the edges of $\Delta _n$ . For $U_{[n]}=\bigoplus _{i \in \mathbb {Z}_n} U(i;n)$ and B being the standard basis for each copy of $\mathbb {C}^n$ , the coefficient quiver $Q(U_{[n]},B)$ has exactly n predecessor free points. This implies the claim since $\Delta _n$ has n arrows. Hence, we can choose at most $2n$ independent parameters.▪

This corollary can also be obtained using the connection of $X(k,n)$ with the affine flag varieties, following Knutson–Lam–Speyer [Reference KnutsonKn08].

5.2 Skeletal torus action

In [Reference Lanini and PützLP20], we introduced an action of $T=(\mathbb {C}^*)^{n+1}$ on quiver Grassmannians for nilpotent representations of the equioriented cycle. We recall the action in the case of our distinguished quiver Grassmannian $X(k,n)$ . First of all, we enumerate the connected components (i.e., segments) of the coefficient quiver and, hence, denote by $s_j$ the (unique) segment ending in j, that is corresponding to the indecomposable summand $U(j;n)$ . We denote moreover by $b_{j,p}$ the basis vector of $U(j;n)$ corresponding to the pth vertex of $s_j$ (e.g., the starting point of $s_j$ is denoted by $b_{j,1}$ , whereas the end vertex is $b_{j,n}$ ). We define a T action on the underlying vector space to M by setting

(5.1) $$ \begin{align} (\gamma_0, \gamma_1,\ldots,\gamma_n).b_{j,p}=\gamma_0^p\gamma_j b_{j,p} \qquad (\gamma_0,\gamma_1, \ldots, \gamma_n)\in T,\quad j,p\in [n] \end{align} $$

and then extending by linearity. By [Reference Lanini and PützLP20, Lemma 5.10], this induces an action on $X(k,n)$ . From now on, we deal with this torus action.

We recall here the definition of skeletality, which is necessary in the next section in order to consider moment graphs.

The following proposition summarizes some results of [Reference Lanini and PützLP20].

6.1 The description

If a complex algebraic variety X is acted upon by a torus T via a skeletal action, one can consider the corresponding moment graph. This is usually an unoriented graph, but if X admits a T-stable cellular decomposition (as in our case), one can give the edges an orientation. More precisely, we have the following definition.

The label of any edge is only well defined up to a sign, but since this does not play any role in the applications (e.g., computation of equivariant cohomology), we assume that the labels are fixed once and for all, and forget about this ambiguity.

We want to explicitly describe the moment graph corresponding to the torus action on $X(k,n)$ from Section 5.2.

First of all, we need to relate the second basis of Definition 4.3 to the basis $\{b_{j,p}\}_{j,p\in {\mathbb Z}_n}$ of $U_{[n]}$ that we used to define the T action (5.1). It is immediate to see (by induction on $n-p$ ) that

$$\begin{align*}v_p^{(j+p)}=b_{j,p}, \quad \mbox{for all } j,p\in {\mathbb Z}_n. \end{align*}$$

For the rest of this section, we consider $B=\bigcup _{i\in {\mathbb Z}_n}\{v^{(i)}_1, \ldots , v_n^{(i)}\}$ . Thus, a successor closed subquiver $Q'$ of dimension $(k, \ldots , k)$ of $Q(U_{[n]},B)$ is a full subquiver whose vertex set is $Q^{\prime }_0=Q^{\prime (1)}_0\bigsqcup Q^{\prime (2)}_0\bigsqcup \cdots \bigsqcup Q^{\prime (n)}_0$ , with $Q^{\prime (i)}_0=\{v^{(i)}_{h_1},\ldots , v^{(i)}_{h_k}\}=Q^{\prime }_0\cap B^{(i)}$ such that for any $i\in [n]$ ,

$$\begin{align*}Q^{\prime(i+1)}_0=\left\{ \begin{array}{ll} \left\{v^{(i+1)}_{h_1+1},\ldots, v^{(i+1)}_{h_k+1}\right\}, &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\mbox{if }v_n^{(i)}\not\in Q^{\prime(i)}_0,\\ \left\{v^{(i+1)}_{h_j+1} \Big\vert v^{(i)}_{h_j}\in Q^{\prime(i)}_0,\ h_j\neq n\right\}\cup\left\{ v_h^{(i+1)} \right\} \mathrm{for some }h\in[n],& \mbox{otherwise}. \end{array} \right. \end{align*}$$

By Theorem 4.10, Proposition 5.5, and [Reference Cerulli IrelliCI11, Proposition 1], both Grassmann necklaces and successor closed subquivers of $Q(U_{[n]},B)$ of dimension $(k,k,\ldots ,k)$ parametrize the T fixed point set. We write down here the explicit correspondence

(6.1) $$ \begin{align} \psi:\textrm{SC}_{\mathbf{k}}(U_{[n]})\rightarrow \mathcal{GN}_{k,n}\qquad Q'\mapsto \left(\psi^{(a)}(Q^{\prime(a)}_0)\right)_{a\in [n]}, \end{align} $$

where $\psi ^{(a)}(Q^{\prime (a)}_0)\in {[n]\choose k}$ is the image of $Q_0^{\prime (a)}$ under the following map:

$$\begin{align*}\psi^{(a)}:B^{(a)}\rightarrow [n] \quad v^{(a)}_h\mapsto a-h. \end{align*}$$

It is easy to check that $\psi $ is well defined and bijective.

Recall from Section 2.2 that, for any $a\in [n]$ , there is a corresponding total order $\leq _a$ on $[n]$ . The corresponding partial order on $\mathcal {GN}_{k,n}$ was denoted by $\leq $ . For any $a,x,y\in [n]$ , we define

$$\begin{align*}[x,y]_a=\{z\in [n]\mid x\leq_a z\leq_a y \}, \end{align*}$$

and $[x,y)_a=[x,y]_a\setminus \{y\}$ .

The following notion allows us to provide a combinatorial description of the moment graph.

We are now ready to provide a description of the moment graph of the T-action on $X(k,n)$ which only involves Grassmann necklace combinatorics.

Proof By [Reference Lanini and PützLP20, Theorem 6.13], $\mathcal {G}_0$ can be identified with $\textrm {SC}_{\mathbf {k}}(U_{[n]})$ , which in turn we identify with $\mathcal {GN}_{k,n}$ via the bijection $\psi $ . Again, by [Reference Lanini and PützLP20, Theorem 6.13], there is an edge $Q'\rightarrow Q"$ for $Q',Q"\in \textrm {SC}_{\mathbf {k}}(M)$ if and only if $Q"$ is obtained from $Q'$ by a mutation, that is, there exist $a,p,p',l\in [n]$ with $p<p'$ such that:

• • $Q'\setminus (Q"\cap Q')=v^{(a)}_p\to v^{(a+1)}_{p+1}\to \cdots \to v^{(a+l-1)}_{p+l-1}. $

• • $Q"\setminus (Q'\cap Q")=v^{(a)}_{p'}\to v^{(a+1)}_{p'+1}\to \cdots \to v^{(a+l-1)}_{p'+l-1}$ .

Since $v_p^{(a)}\in s_{a-p}$ and $v_p^{\prime (a)}\in s_{a-p'}$ , then

$$\begin{align*}Q'\setminus (Q"\cap Q')\subseteq s_{a-p}, \quad Q"\setminus (Q"\cap Q')\subseteq s_{a-p'}.\end{align*}$$

If we set $j:=a-p$ and $j':=a-p'$ , the corresponding edge is labeled by $\epsilon _j-\epsilon _{j'}+(p-p')\delta $ .

Now, we want to translate the conditions on $Q',Q"$ into conditions on the pair of Grassmann necklaces $\psi (Q')=\mathcal {I},\psi (Q")=\mathcal {I}'$ . Observe that $j'<_a j$ as $p'>p$ , and set $b:=a+l-1$ . Hence, we have that the above conditions on Q and $Q'$ translate into:

• • $I_h=I^{\prime }_h$ for all $h\not \in \{a,a+1,\ldots ,a+l-1\}=[a,b]_{a}$ ,

• • $I_h\neq I^{\prime }_h$ and $I_h=(I_h'\setminus \{j'\})\cup \{j\}$ for all $h\in [a,b]_a$ ,

where the second condition follows from the fact that

$$\begin{align*}\psi^{(a)}(v^{(a)}_p)=\psi^{(a+1)}(v^{(a+1)}_{p+1})=\psi^{(a+l-1)}(v^{(a+l-1)}_{p+l-1})=a-p=j\end{align*}$$

and

$$\begin{align*}\psi^{(a)}(v^{(a)}_{p'})=\psi^{(a+1)}(v^{(a+1)}_{p'+1})=\psi^{(a+l-1)}(v^{(a+l-1)}_{p'+l-1})=a-p'=j'.\end{align*}$$

Notice that $j'<_hj$ for any $h\in [a,b]_a$ , and thus $I^{\prime }_h\leq _h I_h$ for any $h\in [n]$ , that is, $I'<I$ . Finally, observe that $p-p'=-\#[j',j)_a$ .

Example 6.5 Let $n=3$ and $k=1$ . We keep the same notation as in Example 2.7 and write $i_1i_2i_3$ for the Grassmann necklace $(I_1=\{i_1\},I_2=\{i_2\},I_3=\{i_3\})\in \mathcal {GN}_{1,3}$ . Moreover, to shorten notation, we write $\alpha _{i,j}:=\epsilon _i-\epsilon _j$ for $i,j\in [3]$ . Then, by Proposition 6.3, the corresponding moment graph is

Notice that the underlying graph coincides with the Hasse diagram we found in Example 2.7. This if of course not always the case, as the Hasse diagram is in general missing several of the edges of the moment graph.

6.2 T-equivariant cohomology

By [Reference Lanini and PützLP20, Corollary 5.13], the moment graph described in the previous result encodes all needed information to compute the (T-equivariant) cohomology ring of $X(k,n)$ . More precisely, let $R:=\mathbb {Q}[\epsilon _1, \ldots , \epsilon _n, \delta ]$ and consider it as a $\mathbb {Z}$ -graded ring with $\deg (\epsilon _i)=\deg (\delta )=2$ ( $i\in [n]$ ). Denote by $\alpha (\mathcal {I},\mathcal {I}')$ the label of the edge $\mathcal {I}\to \mathcal {I}'$ . Corollary 5.13 of [Reference Lanini and PützLP20] gives immediately the following result.

Corollary 6.6 There is an isomorphism of ( $\mathbb {Z}$ -graded) rings:

$$\begin{align*}H_T^*(X(k,n),\mathbb{Q})\simeq \left\{(z_{\mathcal{I}})\in\bigoplus_{\mathcal{I}\in\mathcal{GN}_{k,n}}R \, \Bigg\vert \begin{array}{c} z_{\mathcal{I}}\equiv z_{\mathcal{I}'}\quad\mod \alpha(\mathcal{I},\mathcal{I}')\\ \forall \ \mathrm{edge} \ \mathcal{I}\rightarrow\mathcal{I}' \end{array} \right\}. \end{align*}$$

Moreover, by [Reference Lanini and PützLP21, Theorem 3.21], $H_T^\bullet (X(k,n))$ admits a very nice basis as a free module over R, namely a so-called Knutson–Tao basis (cf. [Reference Lanini and PützLP21, Definition 3.2]).

Example 6.7 By Corollary 6.6, the T-equivariant cohomology of $X(1,3)$ can be read off from the moment graph from Example 6.5:

$$\begin{align*}H_T^*(X(1,3))\simeq \left\{(z_{123},z_{121}, z_{133},z_{223},z_{111},z_{222},z_{333}) \,\Bigg\vert \begin{array}{c} z_{i_1i_2i_3}\equiv z_{i_1'i_2'i_3'}\\\quad\mod \alpha(i_1i_2i_3,i_1'i_2'i_3')\\ \forall \mbox{ edge } \ i_1i_2i_3\rightarrow i_1'i_2'i_3' \end{array} \right\}, \end{align*}$$

where all $z_{i_1i_2i_3}\in R=\mathbb {Q}[\epsilon _1,\epsilon _2,\epsilon _3,\delta ]$ . In this case, the Knutson–Tao basis is

$$\begin{align*}(1,1,1,1,1,1,1), \ (0,0,\alpha_{3,2}-\delta,0,\alpha_{1,2}-2\delta,0,\alpha_{3,2}-\delta), \end{align*}$$

$$\begin{align*}(0,\alpha_{1,3}-\delta,0,0,\alpha_{1,3}-\delta,\alpha_{2,3}-2\delta,0), \ (0,0,0,\alpha_{2,1}-\delta,0,\alpha_{2,1}-\delta,\alpha_{3,1}-2\delta), \end{align*}$$

$$\begin{align*}(0,0,0,0,(\alpha_{1,2}-2\delta)(\alpha_{1,3}-\delta),0,0),\ (0,0,0,0,0,(\alpha_{2,1}-\delta)(\alpha_{2,3}-2\delta),0), \end{align*}$$

$$\begin{align*}(0,0,0,0,0,0,(\alpha_{2,1}-\delta)(\alpha_{3,2}-\delta)). \end{align*}$$

8.1 The extended cyclic quiver and extended representations

Let us denote by $\widetilde \Delta _n$ the extended cyclic quiver defined by the following data:

• • The vertices of $\widetilde \Delta _n$ are of the form $(i,r)$ with $i\in {\mathbb Z}_n$ , $r=1,\dots ,n$ .

• • There are two types of arrows: $\alpha _{i,r}:(i,r)\to (i+1,r+1)$ , $r<n$ and $\beta _{i,r}: (i,r)\to (i,r-1)$ , $r>1$ .

Now, let $M=((M^{(i)})_{i\in \mathbb {Z}_n}, (M_{\alpha _i})_{i\in {\mathbb Z}_n})$ be a $\Delta _n$ representation. We define the extended $\widetilde \Delta _n$ module $\widetilde M$ as follows. Let $M^{(i,1)}=M^{(i)}$ for $i\in \mathbb {Z}_n$ , and let

$$\begin{align*}\widetilde M^{(i,r)}= M_{\alpha_{i-1}}\dots M_{\alpha_{i-r+1}} M^{(i-r+1)} \qquad (r=2,\dots,n). \end{align*}$$

In particular, $\widetilde M^{(i,r)}\subset M^{(i)}$ for all r. We also have natural surjections and inclusions

(8.1) $$ \begin{align} \widetilde M^{(i,n)}\twoheadleftarrow \widetilde M^{(i-1,n-1)}\twoheadleftarrow \cdots\twoheadleftarrow \widetilde M^{(i-n+1,1)} = M^{(i-n+1)}, \end{align} $$

(8.2) $$ \begin{align} \widetilde M^{(i,n)}\subset \widetilde M^{(i,n-1)} \subset \cdots\subset \widetilde M^{(i,1)} = M^{(i)}. \end{align} $$

We complete the definition of the extended representation $\widetilde M$ by letting

$$\begin{align*}\widetilde M_{\alpha_{i,r}}: \widetilde M^{(i,r)} \to \widetilde M^{(i+1,r+1)},\quad \widetilde M_{\beta_{i,r}}: \widetilde M^{(i,r)} \to \widetilde M^{(i,r-1)} \end{align*}$$

to be the maps from (8.1) and (8.2), respectively.

Example 8.3 Let $n=2$ , and let $M=U(1;2)\oplus U(2;2)\in \mathrm {rep}_{\mathbb {C}}(\Delta _2)$ . Then the $\widetilde \Delta _2$ module $\widetilde M$ looks as follows:

Example 8.4 Let $M=U_{[n]}=\bigoplus _{i=1}^n U(i;n)$ , so that $\dim _{\mathbb C} \widetilde M^{(i,r)}=n-r+1$ . To describe the maps between the vector spaces $\widetilde M^{(i,r)}$ , we use the second basis from Definition 4.3 Then $\widetilde M^{(i,r)}$ is identified with the linear span of vectors $v^{(i)}_h$ with $h=r, \dots ,n$ , and the maps $\widetilde M_{\alpha _{i,r}}$ and $\widetilde M_{\beta _{i,r}}$ are given by

$$\begin{align*}\widetilde M_{\beta_{i,r}} v^{(i)}_h = v^{(i)}_h,\ \widetilde M_{\alpha_{i,r}} v^{(i)}_h = v^{(i+1)}_{h+1}, \text{ if } h<n,\ \widetilde M_{\alpha_{i,r}} v^{(i)}_n =0. \end{align*}$$

We say that an element of the path algebra of $\widetilde \Delta _n$ is an $\alpha $ path, respectively, $\beta $ path, if it is given by the concatenation of maps exclusively of type $\widetilde M_{\alpha _{\bullet }}$ , respectively, $\widetilde M_{\beta _\bullet }$ .

8.2 Construction of the desingularization

Lemma 8.7 Let $M\in \mathrm {rep}_{\mathbb {C}}(\Delta _n)$ , and let $U\subset M$ be an $\mathbf {e}$ -dimensional subrepresentation. Then there exists a natural map

(8.3) $$ \begin{align} \Psi: \mathrm{Gr}_{\mathbf{dim} \, \widetilde U}(\widetilde M) \to \mathrm{Gr}_{\mathbf{e}}(M), \end{align} $$

defined by the following rule: for $N\in \mathrm {Gr}_{\mathbf {dim} \, \widetilde U}(\widetilde M)$ , its image $\Psi (N)\in {\overline S_U}$ is given by

$$\begin{align*}\Psi(N)^{(i)}=N^{(i,1)}, \ \Psi(N)_{\alpha_i} = N_{\beta_{i+1,2}}N_{\alpha_{i,1}}. \end{align*}$$

Recall that the irreducible components of $X(k,n)$ are labeled by the k-element subsets $J\subset \{1,\dots ,n\}$ (see Remark 4.12). We denote the corresponding irreducible component by $X_J(k,n)$ . It contains a point $p_J$ which is a summandwise embedding of the representation

$$\begin{align*}U_J=\bigoplus_{j\in J} U(j;n) \end{align*}$$

into $U_{[n]}=\bigoplus _{i=1}^n U(i;n)$ (special case of [Reference PützPue20, Lemma 4.10]).

The component $X_J(k,n)$ is the closure of the $\mathrm {Aut}_{\Delta _n}(U_{[n]})$ orbit $C(p_J)$ of the point $p_J$ . Each point of the cell $C(p_J)$ is isomorphic to $U_J$ as a $\Delta _n$ module, and points in different cells are not isomorphic. Moreover, each cell contains exactly one $\mathbb {C}^*$ fixed point (see Theorem 4.14).

Now, let us consider the $\widetilde \Delta _n$ module $\widetilde U_J$ , and let $\mathbf {d}_J=\mathbf {dim} \, \widetilde U_J$ , so that $\mathbf {d}_J$ is a vector with components $d_J^{(i,r)}$ .

Lemma 8.8 The numbers $d_J^{(i,r)}$ are explicitly given by

$$\begin{align*}d_J^{(i,r)} = \# \left(J\cap \{i':\ i\le i'\le i+n-r\}\right). \end{align*}$$

In particular, $d_J^{(i,r)}\geq d_J^{(i,r+1)}$ for any $i\in {\mathbb Z}_n$ and $r\in [n-1]$ .

Proof By definition,

$$\begin{align*}d_J^{(i,r)} = \dim_{\mathbb C} U_{J,\alpha_{i-1}}\dots U_{J,\alpha_{i-r+1}} U_J^{(i-r+1)}. \end{align*}$$

One easily sees that

$$\begin{align*}U_{J,\alpha_{i-1}}\dots U_{J,\alpha_{i-r+1}} U(i';n)^{(i-r+1)} \end{align*}$$

is nontrivial (i.e., one-dimensional) if and only if $i\le i'\le i+n-r$ .▪

Proof We show that $\mathrm {Gr}_{\mathbf {d}_J} (\widetilde U_{[n]})$ is isomorphic to a tower of fibrations

$$\begin{align*}\mathrm{Gr}_{\mathbf{d}_J} (\widetilde U_{[n]})=Y_1\to Y_2\to\cdots\to Y_n=\mathrm{pt} \end{align*}$$

such that each map $Y_r\to Y_{r+1}$ , $r=1,\dots ,n-1$ is a fibration with the fiber being a product of classical Grassmann varieties.

A point N of $\mathrm {Gr}_{\mathbf {d}_J} (\widetilde U_{[n]})$ is a collection of subspaces $N^{(i,r)}\subset \widetilde U_{[n]}^{(i,r)}$ . We define $Y_r$ as the image of the natural projection map

$$\begin{align*}\mathrm{Gr}_{\mathbf{d}_J} (\widetilde U_{[n]})\to \prod_{r'=r}^n \prod_{i\in{\mathbb Z}_n} \mathrm{Gr}_{d_J^{(i,r')}}(\widetilde U_{[n]}^{(i,r')}), \quad N\mapsto (N^{(i,r')})_{\substack{i\in{\mathbb Z}_n\\r\leq r'\leq n}}. \end{align*}$$

We start with $r=n$ and then proceed by decreasing induction on r. One has $\dim _{\mathbb C} \widetilde {U}_{[n]}^{(i,n)}=1$ , and $d_J^{(i,n)}$ are either $0$ or $1$ . Thus, $Y_n$ is a product of n points (corresponding to $i\in {\mathbb Z}_n$ ). We also note that

$$\begin{align*}U_{\alpha_{i,n-1}}U_{\beta_{i,n}} (N^{(i,n)}) \subset N^{(i+1,n)}, \end{align*}$$

because the composition $U_{\alpha _{i,n-1}}U_{\beta _{i,n}}$ vanishes on $\widetilde U_{[n]}$ . Here and below, we use the notation $U_\gamma =\widetilde {U}_{[n],\gamma }$ for the map corresponding to the edge $\gamma $ of $\widetilde \Delta _n$ .

Now, assume that all the subspaces $N^{(i,r')}$ are fixed for $r'>r$ . Since N is a subrepresentation, the subspace $N^{(i,r)}\subset \widetilde {U}_{[n]}^{(i,r)}$ has to satisfy the conditions

$$\begin{align*}U_{\beta_{i,r+1}} N^{(i,r+1)}\subset N^{(i,r)},\ U_{\alpha_{i,r}} N^{(i,r)} \subset N^{(i+1,r+1)} \end{align*}$$

and hence

(8.4) $$ \begin{align} U_{\alpha_{i,r}} U_{\beta_{i,r+1}} (N^{(i,r+1)}) \subset N^{(i+1,r+1)}. \end{align} $$

Since $U_{\beta _{i,r+1}}$ is an embedding, $U_{\alpha _{i,r}}$ is a surjection, and condition (8.4) holds, the choice of $N^{(i,r)}$ as above is equivalent to the choice of a point in the Grassmannian

$$\begin{align*}\mathrm{Gr}_{d_J^{(i,r)}-d_J^{(i,r+1)}}\bigl(\widetilde U_{[n]}^{(i,r)}/U_{\beta_{i,r+1}}(N^{(i,r+1)})\bigr). \end{align*}$$

Proof Let us take a point $M\in C(p_J)$ . By Theorem 4.14(i), all the points in the cell are isomorphic as $\Delta _n$ modules, and it follows that

$$\begin{align*}\dim_{\mathbb C} M_{\alpha_{i-r+1}}\dots M_{\alpha_i}M^{(i)} = d_J^{(i,r)} \end{align*}$$

for all vertices $(i,r)$ . Hence, the preimage of M is a single point. Moreover, we claim that $f_J^{-1}(C(p_J))$ is open in $\mathrm {Gr}_{\mathbf { d}_J}(\widetilde M)$ . The proof is analogous to the proof of Theorem 8.10: we consider the tower

$$\begin{align*}f_J^{-1}(C(p_J))=Y_1\cap f_J^{-1}(C(p_J))\to Y_2\cap f_J^{-1}(C(p_J))\to\cdots\to Y_n\cap f_J^{-1}(C(p_J))=\mathrm{pt} \end{align*}$$

and show by decreasing induction on r that at each step one obtains an open part of $Y_r$ .

Recall that by definition $X_J(k,n)$ coincides with the closure of the cell $C(p_J)$ . Therefore,

$$\begin{align*}f_J\bigl(\mathrm{Gr}_{\mathbf{d}_J} (\widetilde U_{[n]})\bigr) = \overline{f_J(f_J^{-1}C(p_J))}=\overline{C(p_J)}=X_J(k,n), \end{align*}$$

where the first equality is true since $f_J^{-1}(C(p_J))$ is open in $\mathrm {Gr}_{\mathbf {d}_J}(\widetilde M)$ .▪

We summarize the whole picture in the following corollary.

8.3 Example

We work out two examples of the desingularization above for $n=4$ and $k=2$ . The variety $X(2,4)$ has six irreducible components, labeled by the cardinality two sets $J\subset [4]$ . Essentially, there are two different cases (up to rotation): $J=\{1,2\}$ and $J=\{1,3\}$ . The quiver $\widetilde \Delta _4$ looks as follows:

Now, let $J=\{1,2\}$ . Then the dimension vector $d_J^{(i,r)}$ is given by

Hence, the desingularization $\mathrm {Gr}_{\mathbf {d}_J} (\widetilde U_{[n]})$ is the tower bundle

(8.5) $$ \begin{align} \mathrm{Gr}_{\mathbf{d}_J} (\widetilde U_{[n]}) = Y_1 \to Y_2 \to Y_3 \to Y_4=pt, \end{align} $$

where the fiber of the fibration $Y_3\to Y_4$ is ${\mathbb P}^1$ (the other three factors are points), the fiber of the fibration $Y_2\to Y_3$ is ${\mathbb P}^1\times {\mathbb P}^1$ (with two trivial factors), and the fiber of the fibration $Y_1\to Y_2$ is ${\mathbb P}^1$ (with three trivial factors).

Now, let $J=\{1,3\}$ . Then the dimension vector $d_J^{(i,r)}$ is given by

The tower (8.5) is as follows: $Y_4$ is a point, the fiber of the fibration $Y_3\to Y_4$ is ${\mathbb P}^1\times {\mathbb P}^1$ (with two trivial factors), the fiber of the fibration $Y_2\to Y_3$ is a point (the product of four points), and the fiber of the fibration $Y_1\to Y_2$ is ${\mathbb P}^1\times {\mathbb P}^1$ (with two trivial factors).

8.4 Geometric and combinatorial properties of the desingularization

Lemma 8.14 The automorphism group of $ \widetilde U_{[n]}$ satisfies

$$\begin{align*}\mathrm{Aut}_{\widetilde \Delta_n}\big( \widetilde U_{[n]}\big) \cong \mathrm{Aut}_{\Delta_n}\big(U_{[n]}\big). \end{align*}$$

Observe that the $\mathrm {G}_{\mathbf {n}}$ action on $\mathrm {R}_{\mathbf {n}}(\Delta _n)$ preserves the relations satisfied by the representations and the automorphism group of $M \in \mathrm {R}_{\mathbf {n}}(\Delta _n)$ is its $\mathrm {G}_{\mathbf {n}}$ stabilizer. Hence, on both sides, we obtain the same groups if we view $U_{[n]}$ and $\widetilde U_{[n]}$ as bounded quiver representations.