2. Background

Let $\Lambda$ be a finite-dimensional basic algebra and denote by ${\mathop{\textrm{mod}}\nolimits }\, \Lambda$ the category of finite-dimensional left $\Lambda$ -modules. We let $\tau$ denote the Auslander-Reiten translate on ${\mathop{\textrm{mod}}\nolimits }\, \Lambda$ . We assume any subcategories $\mathcal{X}$ to be full and closed under isomorphism; we define $\mathcal{X}^{\perp } = \{Y \in{\mathop{\textrm{mod}}\nolimits }\, \Lambda \mid \mathop{\textrm{Hom}}\nolimits\! (\mathcal{X}, Y) = 0\}$ and define $^{\perp }\mathcal{X}$ dually.

Consider $\mathcal{C}({\mathop{\textrm{mod}}\nolimits }\, \Lambda ) ={\mathop{\textrm{mod}}\nolimits }\, \Lambda \amalg{\mathop{\textrm{mod}}\nolimits }\, \Lambda [1]$ as a full subcategory of the bounded derived category $D^b({\mathop{\textrm{mod}}\nolimits }\, \Lambda )$ . For an indecomposable object $U$ in $\mathcal{C}({\mathop{\textrm{mod}}\nolimits }\, \Lambda )$ , we set $\left \lvert U\right \rvert = U$ if $U$ is in ${\mathop{\textrm{mod}}\nolimits }\, \Lambda$ and $\left \lvert U\right \rvert = U[\!-\!1]$ if $U$ is in ${\mathop{\textrm{mod}}\nolimits }\, \Lambda [1]$ . If $U$ in $\mathcal{C}({\mathop{\textrm{mod}}\nolimits }\,\Lambda )$ is basic, we denote by $\mathop{\textrm{rk}}\nolimits\! (U)$ the number of indecomposable summands of $U$ .

We recall some notions from [Reference Adachi, Iyama and Reiten2, §0] (in some cases stated slightly differently, but equivalently). A $\Lambda$ -module $M$ is called $\tau$ -rigid if $\mathop{\textrm{Hom}}\nolimits\! (M,\tau M) = 0$ . A (usually assumed basic) object $M \amalg P[1]$ in $\mathcal{C}({\mathop{\textrm{mod}}\nolimits }\, \Lambda )$ is said to be a support $\tau$ -rigid object if $M$ is a $\tau$ -rigid $\Lambda$ -module and $P$ is a projective $\Lambda$ -module with $\mathop{\textrm{Hom}}\nolimits _\Lambda\! (P, M)= 0$ . An object $U = M \amalg P[1]$ is said to be a support $\tau$ -tilting object if $\mathop{\textrm{rk}}\nolimits\! (U) =\mathop{\textrm{rk}}\nolimits\!(\Lambda )$ . If, in addition, $P=0$ , $U$ is said to be a $\tau$ -tilting module.

Recall that a subcategory $\mathsf{W}$ of ${\mathop{\textrm{mod}}\nolimits }\,\Lambda$ is said to be wide if it is closed under kernels, cokernels and extensions. If a wide subcategory $\mathsf{W}$ of ${\mathop{\textrm{mod}}\nolimits }\, \Lambda$ is equivalent to a module category ${\mathop{\textrm{mod}}\nolimits }\, \Lambda ^{\prime}$ , we set $\mathop{\textrm{rk}}\nolimits \mathsf{W} \,:\!= \mathop{\textrm{rk}}\nolimits \Lambda ^{\prime}$ .

Objects which are $\tau$ -rigid give rise to a particular class of wide subcategories.

Definition 2.1. ([Reference Jasso19, Defn. 3.3]). For a support $\tau$ -rigid object $U = M \amalg P[1]$ in $\mathcal{C}({\mathop{\textrm{mod}}\nolimits }\, \Lambda )$ the category

\begin{equation*}J(U) = (M \amalg P)^{\perp } \cap {^\perp {(\tau M)}}\end{equation*}

is called a $\tau$ -perpendicular subcategory.

In the following Theorem, (a) is from [Reference Demonet, Iyama, Reading, Reiten and Thomas10, Thm. 4.12], [Reference Brüstle, Smith and Treffinger4, Cor. 3.22] and (c) is from [Reference Jasso19, Thm. 3.8]. For (b), see [Reference Enomoto11, Prop. 4.12] and [Reference Enomoto and Sakai12, Lemma 4.7].

Let $\mathsf{W}$ be a $\tau$ -perpendicular subcategory of ${\mathop{\textrm{mod}}\nolimits }\, \Lambda$ . Since $\mathsf{W}$ is equivalent to a module category, we can also consider the $\tau$ -tilting theory of $\mathsf{W}$ . Let $\mathcal{C}(\mathsf{W}) = \mathsf{W} \amalg \mathsf{W}[1]$ , as a subcategory of $D^b({\mathop{\textrm{mod}}\nolimits }\, \Lambda )$ . Note that since $\mathsf{W}$ is an exact subcategory of ${\mathop{\textrm{mod}}\nolimits }\, \Lambda$ , there is a canonical isomorphism

\begin{equation*}\mathop {\textrm {Hom}}\nolimits _{D^b(\mathsf {W})}(X,Y[1]) \simeq \mathop {\textrm {Hom}}\nolimits _{D^b({\mathop {\textrm {mod}}\nolimits } \Lambda )}(X,Y[1]),\end{equation*}

for modules $X,Y$ in $\mathsf{W}$ , so we can also consider $\mathcal{C}(\mathsf{W})$ as a subcategory of $D^b(\mathsf{W})$ .

Note that in general $\tau _{\mathsf{W}} X \not \simeq \tau X$ for a module $X$ in $\mathsf{W}$ , and hence in general, there exist modules which are $\tau$ -rigid in $\mathsf{W}$ but not $\tau$ -rigid in ${\mathop{\textrm{mod}}\nolimits }\,\Lambda$ (see e.g. [Reference Buan and Marsh6, §1] or the end of Section 6). But we do have the following. For a support $\tau$ -rigid object $V = N \amalg Q[1]$ in $\mathcal{C}(\mathsf{W}) \subseteq \mathcal{C}({\mathop{\textrm{mod}}\nolimits }\, \Lambda )$ , set

\begin{equation*}J_{\mathsf{W}}(V) = (N \amalg Q)^{\perp } \cap {^{\perp } {(\tau _{\mathsf{W}} N)}}\cap \mathsf{W}.\end{equation*}

The following useful Lemma follows from [Reference Auslander and Smalø1, Prop. 5.8] (see also [Reference Adachi, Iyama and Reiten2, Prop. 1.2]).

Proof. By Lemma 2.3, $X$ is $\tau$ -rigid in $\mathsf{W}$ implies $\mathop{\textrm{Ext}}\nolimits ^1\!(X,\mathsf{W} \cap \mathop{\textrm{Gen}}\nolimits X ) = 0$ . Hence, also $\mathop{\textrm{Ext}}\nolimits ^1\!(X,\mathsf{W}^{\prime} \cap \mathop{\textrm{Gen}}\nolimits X ) = 0$ , and applying Lemma 2.3 again we obtain that $X$ is $\tau$ -rigid in $\mathsf{W}^{\prime}$ .

The following bijection is crucial. It was proved in [Reference Buan and Marsh6] and can be seen as a refinement of [Reference Jasso19, Thm. 3.16].

Theorem 2.5 ([Reference Buan and Marsh6, Prop. 5.6]). Let $\mathsf{W}$ be a $\tau$ -perpendicular subcategory of ${\mathop{\textrm{mod}}\nolimits }\,\Lambda$ , and let $U$ be a support $\tau$ -rigid object in $\mathcal{C}(\mathsf{W})$ . Then, there is a bijection $\mathcal{E}^{\mathsf{W}}_{U}$ from

\begin{equation*}\left\{X \in \mathop {\textrm {ind}}\nolimits\! (\mathcal {C}(\mathsf {W})) \mid X \amalg U \textit { support } \tau \textit {-rigid in } \mathcal {C}(\mathsf {W})\right\} \setminus \mathop {\textrm {ind}}\nolimits U \end{equation*}

to

\begin{equation*}\left\{X \in \mathop {\textrm {ind}}\nolimits\! (\mathcal {C}(J_{\mathsf {W}}(U)) \mid \textit {$X$ is support $\tau $-rigid in $\mathcal {C}(J_{\mathsf {W}}(U))$} \right\}.\end{equation*}

We denote the inverse of $\mathcal{E}^{\mathsf{W}}_{U}$ by $\mathcal{F}^{\mathsf{W}}_{U}$ and, when $\mathsf{W} ={\mathop{\textrm{mod}}\nolimits }\, \Lambda$ , we denote the map in Theorem 2.5 and its inverse simply by $\mathcal{E}_{U}$ and $\mathcal{F}_{U}$ .

Using the bijection in Theorem 2.5, the following was proved in [Reference Buan and Marsh7, Thms. 1.4, 1.7] for the $\tau$ -tilting finite case. It was generalised in [Reference Buan and Hanson5, Thms. 6.4, 6.12] to arbitrary finite-dimensional algebras.

We also recall the following.

3. Uniqueness

Let $n$ be the number of simple $\Lambda$ -modules. Recall that a complete $\tau$ -exceptional sequence in ${\mathop{\textrm{mod}}\nolimits }\, \Lambda$ is a sequence $(X_1,X_2,\ldots,X_n)$ of indecomposable $\Lambda$ -modules where $X_n$ is $\tau$ -rigid and $(X_1,\ldots,X_{n-1})$ is a $\tau$ -exceptional sequence in $J(X_n)$ . Moreover, a sequence $(X_1,X_2,\ldots,X_n)$ of indecomposable objects in $\mathcal{C}({\mathop{\textrm{mod}}\nolimits }\, \Lambda )$ is a signed $\tau$ -exceptional sequence, if (i) $X_n$ is either a $\tau$ -rigid module or of the form $P[1]$ for some projective $\Lambda$ -module $P$ and (ii) $(X_1,X_2,\ldots,X_{n-1})$ is a signed $\tau$ -exceptional sequence in $J(\left \lvert X_n\right \rvert )$ . Note that this means that $X_{n-1}$ is either $\tau$ -rigid in $J(\left \lvert X_n\right \rvert )$ (i.e. $\tau$ -rigid in the equivalent module category), or $X_{n-1}= P^{\prime}[1]$ , where $P^{\prime}$ is (relative) projective in $J(\left \lvert X_n\right \rvert )$ , and so on.

Recall that $\Lambda$ is said to be $\tau$ -tilting finite if it only has a finite number of indecomposable $\tau$ -rigid modules. In this section, we shall prove the following uniqueness result for $\tau$ -exceptional sequences over such algebras:

We first recall the following:

We next make the following observation, which holds for all finite-dimensional algebras:

Proof. We first claim that $(A_1, A_2, \dots, A_{n-1}, \left \lvert A_n\right \rvert )$ is a signed $\tau$ -exceptional sequence. If $\left \lvert A_n\right \rvert$ is projective, then $J(\left \lvert A_n\right \rvert [1]) = J(\left \lvert A_n\right \rvert )$ . Hence, the initial claim follows from the definition of signed $\tau$ -exceptional sequences. The same argument gives that also $(A_1, A_2, \dots, A_{n-2}, \left \lvert A_{n-1}\right \rvert\!, \left \lvert A_n\right \rvert )$ is a signed $\tau$ -exceptional sequence, and so on.

It is clear that Lemma 3.3 and Theorem 3.1 (a) imply Theorem 3.1 (b), so it is enough to prove Theorem 3.1 (a).

In the remainder of this section, we will prove Theorem 3.1 (a). So, we assume for the remainder of the section that $\Lambda$ is $\tau$ -tilting finite.

We then have the following:

Proof. For (a), we note that, by Lemma 2.7, $\mathsf{W}^{\prime}$ is a wide subcategory of $\mathsf{W}$ . By Theorem 3.2, $\mathsf{W}^{\prime}$ is of the form $J_{\mathsf{W}}(U)$ for some $\tau$ -rigid object $U$ in $\mathsf{W}$ . Hence, by Theorem 2.2, $\mathop{\textrm{rk}}\nolimits \mathsf{W}^{\prime}=\mathop{\textrm{rk}}\nolimits \mathsf{W}-r\leq \mathop{\textrm{rk}}\nolimits \mathsf{W}$ , where $r$ is the number of non-isomorphic indecomposable direct summands of $U$ .

For (b) suppose, in addition, that $\mathop{\textrm{rk}}\nolimits \mathsf{W}=\mathop{\textrm{rk}}\nolimits \mathsf{W}^{\prime}$ . Then, $r=0$ in the above, so $U=0$ , and we have $\mathsf{W}=\mathsf{W}^{\prime}$ as required.

We give an alternative proof of (a) at the end of this section.

Proof. The result is clear for $n=1$ , so we may assume that $n\geq 2$ . Let $\mathsf{W}_n={\mathop{\textrm{mod}}\nolimits }\, \Lambda$ , $\mathsf{W}_{n-1}= J_{\mathsf{W}_n}(A_n)=J(A_n)$ , $\mathsf{W}_{n-2}= J_{\mathsf{W}_{n-1}}(A_{n-1})$ , $\ldots$ , $\mathsf{W}_1= J_{\mathsf{W}_{2}}(A_2)$ . By Proposition 2.2, we have $\mathop{\textrm{rk}}\nolimits \mathsf{W}_i=i$ for $1\leq i\leq n$ . For $0\leq i\leq n-1$ , let $\mathcal{X}_i=\mathsf{W}_{i+1}\cap J(A^{\prime}_n)$ , so $\mathcal{X}_0\subseteq \mathcal{X}_1\subseteq \cdots \mathcal{X}_{n-1}=J(A^{\prime}_n)$ .

By Proposition 2.2, we have $\mathop{\textrm{rk}}\nolimits J(A_n)=\mathop{\textrm{rk}}\nolimits J(A^{\prime}_n)=n-1$ . We have $\mathcal{X}_{n-2}=J(A_n)\cap J(A^{\prime}_n)$ . Assume, for a contradiction, that $\mathop{\textrm{rk}}\nolimits \mathcal{X}_{n-2}\leq n-2$ . Note that $A_1,\ldots,A_{n-1}\in J(A^{\prime}_n)$ . Fix $1\leq i\leq n-2$ . Since $A_{i+1}\in \mathsf{W}_{i+1}=J_{\mathsf{W}_{i+2}}(A_{i+2})$ , we have $A_{i+1}\in \mathcal{X}_i=\mathsf{W}_{i+1}\cap J(A^{\prime}_n)$ . However, $A_{i+1}\not \in \mathsf{W}_i=J_{\mathsf{W}_{i+1}}(A_{i+1})$ , since $\mathop{\textrm{Hom}}\nolimits\! (A_{i+1},A_{i+1})\not =0$ , so $A_{i+1}\not \in \mathcal{X}_{i-1}=\mathsf{W}_i\cap J(A^{\prime}_n)$ . Since $\mathsf{W}_i\subseteq \mathsf{W}_{i+1}$ we see that $\mathcal{X}_{i-1}\subseteq \mathcal{X}_i$ but $\mathcal{X}_{i-1}\not =\mathcal{X}_i$ . Hence, by Lemma 3.4, we have $\mathop{\textrm{rk}}\nolimits \mathcal{X}_{i-1}\lt \mathop{\textrm{rk}}\nolimits \mathcal{X}_i$ . Since $\mathop{\textrm{rk}}\nolimits \mathcal{X}_{n-2}\leq n-2$ , it follows that $\mathop{\textrm{rk}}\nolimits \mathcal{X}_i\leq i$ for $0\leq i\leq n-2$ .

In particular, this means that $\mathcal{X}_0=\mathsf{W}_1\cap J(A^{\prime}_n)=J_{\mathsf{W}_2}(A_2)\cap J(A^{\prime}_n)$ is zero. But this gives a contradiction, since $0\not =A_1\in \mathcal{X}_0$ . Hence, we must have $\mathop{\textrm{rk}}\nolimits \mathcal{X}_{n-2}\geq n-1$ .

Since $\mathcal{X}_{n-2}\subseteq J(A_n)$ , it follows again from Lemma 3.4 that $\mathop{\textrm{rk}}\nolimits \mathcal{X}_{n-2}\leq \mathop{\textrm{rk}}\nolimits J(A_n)=n-1$ , so $\mathop{\textrm{rk}}\nolimits \mathcal{X}_{n-2}=n-1$ . Since $\mathcal{X}_{n-2}\subseteq J(A_n)$ and $\mathcal{X}_{n-2}\subseteq J(A^{\prime}_n)$ , Lemma 3.4 implies that $J(A_n)=J(A_n)\cap J(A^{\prime}_n)=J(A^{\prime}_n)$ , and hence, $A_n=A^{\prime}_n$ by [Reference Buan and Marsh7, Proposition 10.7].

We note the following:

Proof. This follows using the same argument as in the proof of Lemma 3.5, replacing $J(A_n)$ with $\mathsf{W}$ , $J(A^{\prime}_n)$ with $\mathsf{W}^{\prime}$ and replacing $A_1,A_2,\ldots A_{n-1}$ with $A_1,A_2,\ldots,A_m$ .

We can now complete the proof of Theorem 3.1 (a) and hence the main theorem of this section.

Proof of Theorem 3.1 (a). If $t= n$ , this follows directly from Lemma 3.5. Assume $t \in \{1, \dots, n-1\}$ . Let $W^A_n = J(A_n)$ , and for $j \in \{t, \dots, n-1\}$ define recursively $\mathsf{W}^A_j = J_{\mathsf{W}^A_{j+1}}(A_j)$ . Define similarly $\mathsf{W}^B_n = J(B_n)$ and $\mathsf{W}^B_j = J_{\mathsf{W}^B_{j+1}}(B_j)$ . Then, $\mathsf{W}^A_{t+1} = \mathsf{W}^B_{t+1} \,:\!= \mathsf{W}^{\prime}$ , and $\mathsf{W}^{\prime} \simeq{\mathop{\textrm{mod}}\nolimits }\, \Lambda ^{\prime}$ for a finite-dimensional algebra $\Lambda ^{\prime}$ , and we have that $(A_1, A_2, \dots, A_{t-1}, A_t)$ and $(B_1, B_2, \dots, B_{t-1}, B_t) = (A_1, A_2, \dots, A_{t-1}, B_t)$ are complete exceptional sequences in $\mathsf{W}^{\prime}$ . Hence, we obtain that $A_t = B_t$ by Lemma 3.5.

We note that the uniqueness property of Theorem 3.1 also holds for arbitrary finite-dimensional hereditary algebras, by [Reference Crawley-Boevey8, Lemma 2], and we conjecture that the assumption on $\tau$ -tilting finiteness should not be necessary.

An alternative proof of Lemma 3.4 (a) can be given using the theory of bricks. Recall that a $\Lambda$ -module $M$ is called a brick if $\mathop{\textrm{End}}\nolimits\! (M)$ is a division algebra. A set of isoclasses of pairwise Hom-orthogonal bricks is called a semibrick. Let $C$ be a full subcategory of ${\mathop{\textrm{mod}}\nolimits }\, \Lambda$ . We denote by $T(C)$ the smallest torsion class containing $C$ , by $\mathop{\textrm{Gen}}\nolimits\! (C)$ the collection of modules obtained as quotients of finite direct sums of modules in $C$ , and by $\mathop{\textrm{Filt}}\nolimits\! (C)$ the category of modules with filtrations by modules in $C$ . By the argument in [Reference Marks and Šťovíček21, Lemma 3.1], $T(C)=\mathop{\textrm{Filt}}\nolimits\! (\!\mathop{\textrm{Gen}}\nolimits\! (C))$ . A semibrick $S$ is called left finite [Reference Asai3, Definition 1.2] if $T(S)$ is functorially finite. Let $n_{\Lambda }$ denote the number of isomorphism classes of simple $\Lambda$ modules. We recall:

Proof of Lemma 3.4 (a). Note that, by [Reference Jasso19, Thm. 3.8], $\mathsf{W}\simeq{\mathop{\textrm{mod}}\nolimits }\, \Lambda ^{\prime}$ for some finite-dimensional algebra $\Lambda ^{\prime}$ . Let $S$ be the set of isoclasses of simple objects in $\mathsf{W}^{\prime}$ . Then $S$ is a semibrick in $\mathsf{W}$ . By [Reference Buan and Marsh7, Prop. 4.2(b)], $\Lambda ^{\prime}$ is $\tau$ -tilting finite. Hence, by Proposition 3.9, $S$ is left finite in $\mathsf{W}$ . By Proposition 3.8, $\mathop{\textrm{rk}}\nolimits\! (\mathsf{W}^{\prime})=|S|\leq \mathop{\textrm{rk}}\nolimits\! (\mathsf{W})$ .

4. Transposition

We now return to the general case, where $\Lambda$ is an arbitrary finite-dimensional algebra. A sequence $(T_1, T_2, \dots, T_r)$ of indecomposable objects in $\mathcal{C}({\mathop{\textrm{mod}}\nolimits }\, \Lambda )$ is called an ordered $\tau$ -rigid object if $\amalg _{i=1}^r T_i$ is a $\tau$ -rigid object, and an ordered support $\tau$ -tilting object if $r=n\,:\!= \mathop{\textrm{rk}}\nolimits \Lambda$ . The symmetric group acts on the set of ordered support $\tau$ -tilting objects in a wide subcategory $\mathsf{W}$ of ${\mathop{\textrm{mod}}\nolimits }\, \Lambda$ , by reordering. We recall the following theorem from [Reference Buan and Marsh6].

Theorem 4.1 ([Reference Buan and Marsh6, Thm. 5.4]). For each $\tau$ -perpendicular subcategory $\mathsf{W}$ of ${\mathop{\textrm{mod}}\nolimits }\,\Lambda$ , there are mutually inverse bijections

\begin{equation*}\{\textit {ordered support $\tau $-tilting objects in $\mathsf {W}$ } \}\end{equation*}

\begin{equation*}\psi ^{\mathsf {W}} \downarrow \text { } \uparrow \phi ^{\mathsf {W}}\end{equation*}

\begin{equation*}\{\textit {complete signed $\tau $-exceptional sequences in $\mathsf {W}$} \}\end{equation*}

In the case $\mathsf{W}={\mathop{\textrm{mod}}\nolimits }\,\Lambda$ , we write $\psi$ for $\psi ^{{\mathop{\textrm{mod}}\nolimits }\, \Lambda }$ and $\phi$ for $\phi ^{{\mathop{\textrm{mod}}\nolimits }\, \Lambda }$ . In this section, we will describe the action of the symmetric group on complete signed $\tau$ -exceptional sequences in a $\tau$ -perpendicular subcategory $\mathsf{W}$ of ${\mathop{\textrm{mod}}\nolimits }\,\Lambda$ induced by the bijections above.

Remark 4.2. If $\psi (T_1, T_2, \dots, T_n) = (A_1, \dots, A_n)$ , define (as in the proof of Lemma 3.5 ):

\begin{align*} \mathsf{W}_{n-1} & = J(A_n),\\ \mathsf{W}_{n-2} &= J_{\mathsf{W}_{n-1}}(A_{n-1}), \\ & \, \vdots \\ \mathsf{W}_j & = J_{\mathsf{W}_{j+1}}(A_{j+1}), \\ & \, \vdots \\ \mathsf{W}_1 &= J_{\mathsf{W}_{2}}(A_2). \end{align*}

Then, we have

\begin{align*} A_n &= T_n, \\ A_{n-1} &= \mathcal{E}_{A_n}(T_{n-1}), \\ A_{n-2} &= \mathcal{E}^{\mathsf{W}_{n-1}}_{A_{n-1}} \mathcal{E}_{A_n} (T_{n-2}), \\ & \, \vdots \\ A_{n-j} &= \mathcal{E}_{A_{n-j+1}}^{\mathsf{W}_{n-j+1}} \dots \mathcal{E}^{\mathsf{W}_{n-1}}_{A_{n-1}} \mathcal{E}_{A_n}(T_{n-j}), \\ &\, \vdots \\ A_1 &= \mathcal{E}_{A_2}^{\mathsf{W}_2} \dots \mathcal{E}^{\mathsf{W}_{n-1}}_{A_{n-1}} \mathcal{E}_{A_n}(T_1). \end{align*}

Lemma 4.3. With notation as above, we have

\begin{equation*}\mathsf {W}_{n-i} = J(T_n \amalg T_{n-1} \amalg \dots \amalg T_{n-{i+1}}),\end{equation*}

for $i= 1, \dots, n-1$ .

Proof. This follows from repeated use of Theorem 2.6 (a).

Let $\Lambda$ be an algebra of rank $n$ , and let $\mathcal{T}_o^{\Lambda }$ be the set of ordered basic support $\tau$ -tilting objects in $\mathcal{C}({\mathop{\textrm{mod}}\nolimits }\, \Lambda )$ . There is a natural action of the symmetric group $S_n$ on $T_o^{\Lambda }$ , given by $\pi _i (T_1, \dots, T_n) = (T_1, \dots, T_{i-1}, T_{i+1}, T_i, T_{i+2}, \dots, T_n)$ , where $\pi _i$ denotes the simple transposition $(i\ i+1)$ .

Proof. By applying Theorem 2.6 (b) twice, we obtain

\begin{equation*} \mathcal {E}^{J(C)}_{\mathcal {E}_C(B)}\mathcal {E}_C = \mathcal {E}_{C \amalg B} = \mathcal {E}_{B \amalg C} = \mathcal {E}^{J(B)}_{\mathcal {E}_B(C)}\mathcal {E}_B. \end{equation*}

Proposition 4.6. Let $(T_1, \dots, T_n)$ be an ordered support $\tau$ -tilting object in $\mathcal{C}({\mathop{\textrm{mod}}\nolimits }\, \Lambda )$ and assume that $\psi (T_1, \dots, T_n) = (A_1, \dots, A_n)$ . Then

\begin{equation*}\psi (\pi _{n-1}(T_1, \dots, T_n)) = \left(A_1, \dots A_{n-2}, \mathcal {E}_{\mathcal {F}_{A_{n}} (A_{n-1})}(A_n), \mathcal {F}_{A_n}(A_{n-1})\right).\end{equation*}

Proof. Let $\ (B_1, B_2, \dots, B_n) = \psi (\pi _{n-1}(T_1, \dots, T_n)) = \psi (T_1, \dots, T_{n-2}, T_n, T_{n-1})$ . We need to show that

1. (i) $B_n = \mathcal{F}_{A_n}(A_{n-1})$ ,

2. (ii) $B_{n-1} = \mathcal{E}_{\mathcal{F}_{A_{n}}(A_{n-1})}(A_n)$ , and

3. (iii) $B_j = A_j$ , for $1 \leq j \leq n-2$ .

Note that, by Remark 4.2, we have that $B_n=T_{n-1}$ , $A_n=T_n$ and that $A_{n-1} = \mathcal{E}_{T_n}(T_{n-1})$ . Hence, we have $\mathcal{F}_{T_n}(A_{n-1}) = \mathcal{F}_{T_n}\mathcal{E}_{T_n}(T_{n-1}) = T_{n-1} = B_n$ , which proves claim (i). Moreover, it also follows from Remark 4.2 that $B_{n-1} = \mathcal{E}_{T_{n-1}}(T_n) = \mathcal{E}_{\mathcal{F}_{T_n}(A_{n-1})}(A_n)$ , which proves claim (ii).

It remains to prove that $B_{j} = A_j$ for $j \leq n-2$ . Apply Lemma 4.5, with $B= T_{n-1}$ and $C = T_n$ to obtain that

\begin{equation*}\mathcal {E}^{J(T_n)}_{\mathcal {E}_{T_n}(T_{n-1})}\mathcal {E}_{T_n} = \mathcal {E}^{J(T_{n-1})}_{\mathcal {E}_{T_{n-1}}(T_n)}\mathcal {E}_{T_{n-1}}.\end{equation*}

It now follows directly, from Remark 4.2 and Lemma 4.3, that $B_{j} = A_j$ for $j \leq n-2$ .

Proof of Theorem 4.4. By Proposition 4.6, it follows that both (a) and (b) hold for $i = n-1$ . Assume $i \lt n-1$ . Then, $(A_1, \dots, A_{i-1})$ is a complete signed $\tau$ -exceptional sequence for the $\tau$ -perpendicular subcategory $\mathsf{W}_{i-1}$ , as defined in Remark 4.2. Finally, Proposition 4.6 implies that both (a) and (b) hold also in this case.

5. Mutation

For a fixed positive integer $n$ , consider the group $G_n = \langle \mu _1, \dots, \mu _n \mid \mu _i^2 = e \rangle$ (as in [Reference King and Pressland20, §1]). Let $\Lambda$ be a fixed algebra of rank $n$ .

Mutation of support $\tau$ -tilting objects (as in [Reference Adachi, Iyama and Reiten2, Thm. 2.18]) induces a mutation on the set $\mathcal{T}^{\Lambda }_o$ of ordered basic support $\tau$ -tilting objects in $\mathcal{C}({\mathop{\textrm{mod}}\nolimits }\, \Lambda )$ , which can be regarded as an action of $G_n$ on $\mathcal{T}^{\Lambda }_o$ and hence, via the bijections in Theorem 4.1, an action on the set of complete signed $\tau$ -exceptional sequences.

The following result follows from [Reference Adachi, Iyama and Reiten2, Thm. 2.18].

With $T$ and $T(i)$ as above, we set $\mu _i(T) = T(i)$ . This defines a $G$ -action on $\mathcal{T}^{\Lambda }_o$ . We now describe the corresponding action on the set of complete signed $\tau$ -exceptional sequences.

For a complete signed $\tau$ -exceptional sequence $\mathcal{S} = (A_1, \dots, A_n)$ , let

\begin{equation*}s_1(\mathcal {S}) = (\widetilde {A_1}, A_2, \dots, A_n),\end{equation*}

where

\begin{equation*}\widetilde {A_1} = \begin {cases} A_1[1] & \text { if } A_1 \in {\mathop {\textrm {mod}}\nolimits } \Lambda ; \\ A_1[\!-\!1] & \text { if } A_1 \in {\mathop {\textrm {mod}}\nolimits } \Lambda [1]. \end {cases} \end{equation*}

Moreover, for $j\gt 1$ , let $s_j(\mathcal{S}) = \widetilde{\pi }_{j-1} \widetilde{\pi }_{j-2} \dots \widetilde{\pi _1} s_1 \widetilde{\pi }_1 \dots \widetilde{\pi }_{j-2} \widetilde{\pi }_{j-1}(\mathcal{S})$ .

We make the following observation:

Proof. Let $S$ be the unique simple $\Lambda$ -module, and suppose that $X$ is a non-projective indecomposable $\tau$ -rigid $\Lambda$ -module. Then, $\mathop{\textrm{Ext}}\nolimits ^1\!(X,M)\not =0$ for some $\Lambda$ -module $M$ . Since $M$ must be constructed from $S$ by repeated extensions with $S$ , it follows that $\mathop{\textrm{Ext}}\nolimits ^1\!(X,S)\not =0$ . Since $X$ is also constructed from $S$ by repeated extensions with $S$ , we have that $S$ is a factor of $X$ , so $\mathop{\textrm{Ext}}\nolimits ^1\!(X,\mathop{\textrm{Gen}}\nolimits X)\not =0$ . By Lemma 2.3, $X$ is not $\tau$ -rigid.

We now have:

Proof. Define $W^A_j$ , for $j=1,\ldots,n$ , as in the proof of Theorem 3.1 (a). By repeated application of Theorem 2.2, $W^A_j$ is equivalent to a module category over a finite-dimensional algebra of rank $j-1$ for $j=1,\ldots,n$ . Hence, $\mathsf{W}^A_2$ is equivalent to the module category of a finite-dimensional algebra with a unique simple module. By Lemma 5.2, $\mathsf{W}^A_2$ , regarded as a module category, has a unique indecomposable $\tau$ -rigid module, given by the unique indecomposable projective module. This proves that $s_1(\mathcal{S})$ is a signed $\tau$ -exceptional sequence, and also that $\mathcal{S}$ and $s_1(\mathcal{S})$ are the only signed $\tau$ -exceptional sequences of the form $(X, A_2, \dots, A_n)$ , that is the claim for $j=1$ .

The claim for $j\gt 1$ follows by combining this with Theorem 4.4 (a).

Proof. Consider first the case $i=1$ , and assume that $\psi (M_1, M_2, \dots, M_n) = (A_1, A_2, \dots,A_n)$ . We have $\mu _1(M_1, M_2, \dots, M_n) = (M_1^\ast, M_2, \dots, M_n)$ , where $M^\ast _1 \not \simeq M_1$ , and so $\psi \mu _1(M_1, M_2, \dots, M_n) = \psi (M_1^\ast, M_2, \dots, M_n) =(X, A_2, \dots,A_n)$ , for some object $X$ . The claim for $i=1$ now follows from Lemma 5.3.

For $j\gt 1$ , we first note that $\mu _j = \pi _{j-1} \pi _{j-2} \dots \pi _1 \mu _1 \pi _1\dots \pi _{j-2}\pi _{j-1}$ .

Next we note that, by repeated applications of Theorem 4.4, we have that

\begin{equation*}\widetilde {\pi _1} \dots \widetilde {\pi _j} \psi = \psi \pi _1 \dots \pi _j.\end{equation*}

Combining this with the above, we obtain

\begin{align*} s_j \psi &= (\widetilde{\pi }_{j-1} \widetilde{\pi }_{j-2} \dots \widetilde{\pi }_1 s_1 \widetilde{\pi }_1 \dots \widetilde{\pi }_{j-2} \widetilde{\pi }_{j-1}) \psi \\ &= (\widetilde{\pi }_{j-1} \widetilde{\pi }_{j-2} \dots \widetilde{\pi }_1 s_1) (\widetilde{\pi }_1 \dots \widetilde{\pi }_{j-2} \widetilde{\pi }_{j-1} \psi ) \\ &= (\widetilde{\pi }_{j-1} \widetilde{\pi }_{j-2} \dots \widetilde{\pi }_1 s_1) (\psi \pi _1 \dots \pi _{j-2} \pi _{j-1}) \\ &= (\widetilde{\pi }_{j-1} \widetilde{\pi }_{j-2} \dots \widetilde{\pi }_1 )(s_1 \psi ) (\pi _1 \dots \pi _{j-2} \pi _{j-1}) \\ &= (\widetilde{\pi }_{j-1} \widetilde{\pi }_{j-2} \dots \widetilde{\pi }_1 )(\psi \mu _1) (\pi _1 \dots \pi _{j-2} \pi _{j-1}) \\ &= (\widetilde{\pi }_{j-1} \widetilde{\pi }_{j-2} \dots \widetilde{\pi }_1 \psi ) (\mu _1\pi _1 \dots \pi _{j-2} \pi _{j-1}) \\ & = (\psi \pi _{j-1} \pi _{j-2} \dots \pi _1) (\mu _1 \pi _1\dots \pi _{j-2}\pi _{j-1}) \\ & = \psi (\pi _{j-1} \pi _{j-2} \dots \pi _1\mu _1 \pi _1\dots \pi _{j-2}\pi _{j-1}) \\ &= \psi \mu _j. \end{align*}

King and Pressland [Reference King and Pressland20, Defn. 1.2] consider the following group:

They show that this group acts naturally on labelled (i.e. ordered) seeds in a cluster algebra [Reference Fomin and Zelevinsky13] via permutation and mutation. The mutation of support- $\tau$ -tilting objects in [Reference Adachi, Iyama and Reiten2, §2.3] can be regarded as a generalisation of cluster mutation, so it is natural to consider the action of the mutation group in this context. Note that, for an ordered support $\tau$ -tilting object $T$ , we have $\sigma (\mu _i T) = \mu _{\sigma (i)} (\sigma T)$ for a permutation $\sigma$ and mutation $\mu _i$ , so $M_n$ acts on the set of all ordered support $\tau$ -tilting objects, $\mathcal{T}_o^{\Lambda }$ .

We get an induced action of the mutation group on the set of signed $\tau$ -exceptional sequences.

Theorem 5.6. Let $\mathcal{S} = (A_1, A_2, \dots, A_n)$ be a signed $\tau$ -exceptional sequence. The operations

\begin{equation*}\widetilde {\pi _i}(\mathcal {S})\,:\!= (A_1, \dots, A_{i-1}, \mathcal {E}^{(i)} (A_{i+1}), \mathcal {F}^{(i)}(A_i), A_{i+2}, \dots, A_n),\end{equation*}

\begin{equation*}s_1(\mathcal {S}) = (\widetilde {A_1}, A_2, \dots, A_n),\end{equation*}

and, for $j=2\ldots,n$ ,

\begin{equation*}s_j(\mathcal {S}) = \widetilde {\pi _{j-1}} \widetilde {\pi _{j-2}} \dots \widetilde {\pi _1} s_1 \widetilde {\pi _1} \dots \widetilde {\pi _{j-2}} \widetilde {\pi _{j-1}} (\mathcal {S}),\end{equation*}

define an action of the mutation group $M_n$ on the set of signed $\tau$ -exceptional sequences.

Proof. As already noted, $M_n$ acts on the set of ordered support $\tau$ -tilting objects, and the result hence follows directly from combining Theorem 4.4 and Proposition 5.4 with the fact that $\psi$ is a bijection between the set of ordered support $\tau$ -tilting objects and the set of signed $\tau$ -exceptional sequences (Theorem 4.1).

6. Examples relating to braid actions

Note that the braid group, $B_n$ , on $n$ strands, has the symmetric group $S_n$ as a quotient. Since $S_n$ is a subgroup of the mutation group $M_n=S_n \ltimes G_n$ , it follows that $B_n$ acts naturally on the set of all ordered support $\tau$ -tilting objects $\mathcal{T}_o^{\Lambda }$ and thus on the set of all complete signed $\tau$ -exceptional sequences for $\Lambda$ , by Theorem 4.1. However, this action is highly non-transitive in general, since the braid group is only permuting the possible orderings of each support $\tau$ -tilting object.

It is therefore natural to ask whether there is a transitive action. In the first part of this section, we give an example to show that, at least via the mutation group, this is not possible: we give an algebra for which there is no transitive action of $B_2$ which factors through the action of $M_2$ on $\mathcal{T}_o^{\Lambda }$ .

Let $Q$ be the Kronecker quiver , and let $\Lambda$ be the corresponding path algebra. Let $P_i$ (respectively, $I_i$ ), for $i=1,2$ be the indecomposable projective (respectively, injective) $\Lambda$ -modules corresponding to the vertices of $Q$ . Then, the $\tau$ -tilting (equivalently, tilting) $\Lambda$ -modules are the modules $\tau ^{-r}P_1\amalg \tau ^{-r}P_2$ , $\tau ^{-r}P_1\amalg \tau ^{-(r+1)}P_2$ , $\tau ^r I_1 \amalg \tau ^r I_2$ and $\tau ^{r+1} I_1 \amalg \tau ^rI_2$ , for $r=0,1,2,\ldots$ . The support $\tau$ -tilting (equivalently, support tilting) objects over $\Lambda$ which are not $\tau$ -tilting are $I_1\amalg P_2[1]$ , $P_1[1]\amalg P_2$ and $P_1[1]\amalg P_2[1]$ . In particular, note that $\Lambda$ is not $\tau$ -tilting finite.

The mutation group (see Definition 5.5) is $M_2=S_2\ltimes G_2$ , where $S_2=\{1,\sigma \}$ is the symmetric group of degree $2$ and $G_2=\langle \mu _1,\mu _2 \ :\ \mu _1^2=\mu _2^2=e\rangle$ . We have $\sigma \mu _1=\mu _2\sigma$ and $\sigma \mu _2=\mu _1\sigma$ . The action of $M_2$ on the set $\mathcal{T}_o^{\Lambda }$ of ordered basic support $\tau$ -tilting objects is shown in Figure 1, from which it can be seen that this action is transitive. However, we have the following.

Figure 1. The action of $M_2$ on $T_o^{\Lambda }$ for the Kronecker algebra $\Lambda$ .

Proof. Note that the braid group $B_2$ on two strands is isomorphic to the infinite cyclic group. If there was a transitive action of $B_2$ on $\mathcal{T}_o^{\Lambda }$ factoring through the action of $M_2$ on $\mathcal{T}_o^{\Lambda }$ , then there would be a subgroup of $M_2$ which is a quotient of $B_2$ which acted transitively on $\mathcal{T}_o^{\Lambda }$ . Such a subgroup would have to be cyclic.

The elements of $M_2$ are of the form

\begin{equation*}\sigma ^\varepsilon \mu _i\mu _{\sigma (i)}\mu _i\cdots \mu _i\end{equation*}

and

\begin{equation*}\sigma ^\varepsilon \mu _i\mu _{\sigma (i)}\mu _i\cdots \mu _{\sigma (i)},\end{equation*}

where $i\in \{1,2\}$ and $\varepsilon \in \{0,1\}$ . It is easy to check from the description of the action of $M_2$ (see Figure 1) that, for each of these elements, the (infinite) cyclic group it generates does not act transitively on $\mathcal{T}_o^{\Lambda }$ .

The transitive braid group action on the set of complete exceptional sequences for a hereditary algebra arises in the following way [Reference Crawley-Boevey8, Lemma 9, Theorem], [Reference Ringel23, §5, §7]. Write the braid group $B_n$ on $n$ strands in the usual way as

\begin{equation*}B_n=\left \langle \sigma _1,\ldots,\sigma _{n-1}\,:\,\begin {array}{cc} \sigma _i\sigma _j=\sigma _j\sigma _i, & |i-j|\gt 1; \\ \sigma _i\sigma _j\sigma _i=\sigma _j\sigma _j\sigma _i, & |i-j|=1 \end {array} \right \rangle. \end{equation*}

Given a complete exceptional sequence $(X_1,\ldots,X_n)$ and $1\leq i\leq n$ , there is a unique complete exceptional sequence of the form $(X_1,\ldots,X_{i-1},X_{i+1},Y,X_{i+2},\ldots,X_n)$ for some exceptional indecomposable module $Y$ . The left version of the action of the braid group is given by:

\begin{equation*}\sigma _i(X_1,\ldots,X_n)= (X_1,\ldots,X_{i-1},X_{i+1},Y,X_{i+2},\ldots,X_n).\end{equation*}

If $n=2$ , this means, in particular, that if $(X_1,X_2)$ is a complete exceptional sequence then there is an exceptional sequence of the form $(X_2,Y)$ .

However, if $(X_1,X_2)$ is a complete $\tau$ -exceptional sequence, it can be the case that there is no $\tau$ -exceptional sequence of the form $(X_2,Y)$ , that is it can happen that $X_2\not \in J(Y)$ for all indecomposable $\tau$ -rigid modules $Y$ . We illustrate this point with the following example from [Reference Buan and Marsh6, §6.2].

Let $\Gamma$ be the algebra given by the path algebra of the quiver , subject to the relation $\beta \alpha =0$ . The $\tau$ -exceptional sequences over $\Gamma$ were given in [Reference Buan and Marsh6, §6.2]. They are as follows:

\begin{equation*}\left (1,\begin {matrix} 2 \\ 1 \\ 2\end {matrix}\right ), \left (2,\begin {matrix} 1 \\ 2 \end {matrix}\right ), \left (\begin {matrix} 2 \\ 1 \end {matrix},2\right ), \left (\begin {matrix} 1 \\ 2 \end {matrix},1\right ).\end{equation*}

We make the following observation.

Proof. This can be seen from the list of $\tau$ -exceptional sequences in ${\mathop{\textrm{mod}}\nolimits }\,\Gamma$ above, since the module $P_2=\begin{smallmatrix} 2 \\ 1 \\ 2 \end{smallmatrix}$ , despite being $\tau$ -rigid, does not occur as the first term in any of the complete $\tau$ -exceptional sequences in the list.

Note that it follows that $P_2$ also does not occur as the first term in a complete signed $\tau$ -exceptional sequence.

The phenomenon described above also occurs with the right version [Reference Crawley-Boevey8, Reference Ringel23] of the action of the braid group. Given a complete exceptional sequence $(X_1,\ldots,X_n)$ and $1\leq i\leq n$ , there is a unique complete exceptional sequence of the form $(X_1,\ldots,X_{i-1},Y,X_{i+1},X_{i+2},\ldots,X_n)$ for some exceptional indecomposable module $Y$ . The right version of the action of the braid group is given by:

\begin{equation*}\sigma _i(X_1,\ldots,X_n)= (X_1,\ldots,X_{i-1},Y,X_{i+1},X_{i+2},\ldots,X_n).\end{equation*}

If $n=2$ , this means, in particular, that if $(X_1,X_2)$ is a complete exceptional sequence then there is an exceptional sequence of the form $(Y,X_1)$ .

We can see from the list above that, although $(\begin{smallmatrix} 2 \\ 1 \end{smallmatrix},2)$ is a complete $\tau$ -exceptional sequence for $\Gamma$ , there is no such sequence of the form $(Y,\begin{smallmatrix} 2 \\ 1\end{smallmatrix})$ . This is because, although $\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}$ is $\tau$ -rigid in $J(2)$ , it is not a $\tau$ -rigid $\Gamma$ -module. In the hereditary case, where a module $M$ is $\tau$ -rigid if and only if $\mathop{\textrm{Ext}}\nolimits ^1\!(M,M) = 0$ , a module in $J(Y) = Y^{\perp _{0,1}}$ is $\tau$ -rigid in $J(Y)$ if and only if it is $\tau$ -rigid in ${\mathop{\textrm{mod}}\nolimits }\,\Lambda$ , but, as we see here, this is not true in general.