1.1 Roots and weights

Let $\Phi $ be a simple root system in a Euclidean space $X_{\mathbb {R}}$ with scalar product $(-\,,-)$ . Let $\Phi ^+ \subseteq \Phi $ be a positive system, corresponding to a base $\Pi = \{ \alpha _1, \ldots ,\alpha _n \}$ of $\Phi $ , and write $\alpha _{\mathrm {hs}}$ and $\alpha _{\mathrm {h}}$ for the highest short root and the highest root in $\Phi $ , respectively (with the convention that $\alpha _{\mathrm {hs}} = \alpha _{\mathrm {h}}$ if $\Phi $ is simply laced). For $\alpha \in \Phi $ , let $\alpha ^\vee = 2\alpha /(\alpha ,\alpha )$ be the coroot of $\alpha $ , and let $s_\alpha $ be the reflection corresponding to $\alpha $ , with $s_\alpha (x) = x - (x,\alpha ^\vee ) \cdot \alpha $ for $x \in X_{\mathbb {R}}$ . We write

$$\begin{align*}X = \{ \lambda \in X_{\mathbb{R}} \mid (\lambda,\alpha^\vee) \in \mathbb{Z} \text{ for all } \alpha \in \Phi \} \end{align*}$$

for the weight lattice and $W_{\mathrm {fin}} = \langle s_\alpha \mid \alpha \in \Phi \rangle $ for the (finite) Weyl group of $\Phi $ , with simple reflections $S_{\mathrm {fin}} = \{ s_\alpha \mid \alpha \in \Pi \}$ and length function $\ell \colon W_{\mathrm {fin}} \to \mathbb {Z}_{\geq 0}$ . Furthermore, we write $\rho = \frac 12 \cdot \sum _{\alpha \in \Phi ^+} \alpha $ and denote by $h = (\rho ,\alpha _{\mathrm {hs}}^\vee )+1$ the Coxeter number of $\Phi $ . The set of dominant weights is

$$\begin{align*}X^+ = \{ \lambda \in X \mid (\lambda,\alpha^\vee) \geq 0 \text{ for all } \alpha \in \Phi^+ \} , \end{align*}$$

and the fundamental dominant weights $\{ \varpi _1 , \ldots , \varpi _n \}$ are defined by $(\varpi _i,\alpha _j^\vee ) = \delta _{ij}$ for $1 \leq i,j \leq n$ . We consider the partial order $\leq $ on X such that $\lambda \leq \mu $ if and only if $\mu - \lambda \in \sum _{\alpha \in \Pi } \mathbb {Z}_{\geq 0} \cdot \alpha $ .

1.2 Representations and characters

Let $\Bbbk $ be an algebraically closed field of characteristic $p>0$ and let G be the unique (up to isomorphism) simply connected simple algebraic group over $\Bbbk $ with root system $\Phi $ . We identify X with the character lattice of a fixed maximal torus of G and write $\mathrm {Rep}(G)$ for the category of finite-dimensional rational G-modules. In the following, we will simply refer to the objects of $\mathrm {Rep}(G)$ as G-modules (omitting the mention of rationality and finite-dimensionality).

Every G-module M admits a weight space decomposition $M = \bigoplus _{\mu \in X} M_\mu $ , and the character of M is defined as $\operatorname {\mathrm {ch}} M = \sum _\mu \dim M_\mu \cdot e^\mu \in \mathbb {Z}[X]$ . The simple G-modules are determined up to isomorphism by their highest weight in $X^+$ , and we write $L(\lambda )$ for the simple G-module of highest weight $\lambda $ . Every G-module M has a finite composition series, and we write $[M:L(\lambda )]$ for the multiplicity of the simple G-module $L(\lambda )$ in a composition series of M. We also denote by $\nabla (\lambda )$ and $\Delta (\lambda )$ the induced module and the Weyl module of highest weight $\lambda \in X^+$ (see Subsections II.2.1–2 and II.2.13 in [Reference JantzenJan03]), so that $L(\lambda )$ is the unique simple submodule of $\nabla (\lambda )$ and the unique simple quotient of $\Delta (\lambda )$ . Writing $M^*$ for the dual of a G-module M, we have $L(\lambda )^* \cong L(-w_0\lambda )$ and $\nabla (\lambda )^* \cong \Delta (-w_0\lambda )$ for all $\lambda \in X^+$ , where $w_0 \in W_{\mathrm {fin}}$ denotes the longest element.

The characters of both $\nabla (\lambda )$ and $\Delta (\lambda )$ are given by the well-known Weyl character formula, cf. Subsections II.5.10–11 in [Reference JantzenJan03]. We write $\chi (\lambda ) = \operatorname {\mathrm {ch}} \nabla (\lambda ) = \operatorname {\mathrm {ch}} \Delta (\lambda )$ and extend the definition of $\chi (\lambda )$ to all $\lambda \in X$ (possibly nondominant) by requiring that $\chi (w \boldsymbol {\cdot } \mu ) = (-1)^{\ell (w)} \cdot \chi (\mu )$ for $\mu \in X$ and $w \in W_{\mathrm {fin}}$ , and $\chi (\mu )=0$ if $(\mu ,\alpha ^\vee ) = -1$ for some $\alpha \in \Pi $ . The characters $\chi (\lambda )$ with $\lambda \in X^+$ form a basis of the ring $\mathbb {Z}[X]^{W_{\mathrm {fin}}}$ of $W_{\mathrm {fin}}$ -invariants in $\mathbb {Z}[X]$ , and so do the characters $\operatorname {\mathrm {ch}} L(\lambda )$ of the simple G-modules, by Section II.5.8 in [Reference JantzenJan03]. We write

$$\begin{align*}\Lambda(\lambda) = \{ \mu \in X \mid \Delta(\lambda)_\mu \neq 0 \} \end{align*}$$

for the set of weights of $\Delta (\lambda )$ and note that $\Lambda (\lambda ) = \mathrm {conv}( W_{\mathrm {fin}} \lambda ) \cap ( \lambda + \mathbb {Z}\Phi )$ by [Reference Fulton and HarrisFH91, Theorem 14.18], where $\mathrm {conv}(-)$ denotes the convex hull of a subset of $X_{\mathbb {R}}$ .

A good filtration of a G-module M is a filtration

$$\begin{align*}0 = M_0 \subseteq \cdots \subseteq M_r = M \end{align*}$$

such that $M_i / M_{i-1} \cong \nabla (\lambda _i)$ for some $\lambda _i \in X^+$ for $i=1,\ldots ,r$ , and a Weyl filtration of a G-module N is a filtration

$$\begin{align*}0 = N_0 \subseteq \cdots \subseteq N_s = N \end{align*}$$

such that $N_i / N_{i-1} \cong \Delta (\mu _i)$ for some $\mu _i \in X^+$ for $i=1,\ldots ,s$ . For a G-module M that admits a good filtration (as above), the multiplicity of $\nabla (\lambda )$ in any good filtration of M is defined by

for all $\lambda \in X^+$ (see Proposition II.4.16 in [Reference JantzenJan03]), and for a G-module N that admits a Weyl filtration, the multiplicity $[N : \Delta (\lambda )]_\Delta $ of $\Delta (\lambda )$ in a Weyl filtration of N is defined analogously. A G-module M is called a tilting module if it admits both a good filtration and a Weyl filtration. For all $\lambda \in X^+$ , there is a unique indecomposable tilting module $T(\lambda )$ of highest weight $\lambda $ , and every tilting module can be written as a finite direct sum of these indecomposable tilting modules [Reference RingelRin91, Reference DonkinDon93]. Furthermore, the class of G-modules admitting a good filtration is closed under tensor products [Reference WangWan82, Reference DonkinDon85, Reference MathieuMat90], and hence so are the class of all G-modules admitting a Weyl filtration and the full subcategory $\mathrm {Tilt}(G)$ of tilting modules in $\mathrm {Rep}(G)$ .

1.3 Alcove geometry

Let $W_{\mathrm {aff}} = \mathbb {Z}\Phi \rtimes W_{\mathrm {fin}}$ be the affine Weyl group of G. We write $\gamma \mapsto t_\gamma $ for the canonical embedding of $\mathbb {Z}\Phi $ into $\mathbb {Z}\Phi \rtimes W_{\mathrm {fin}} = W_{\mathrm {aff}}$ . The (p-dilated) dot action of $W_{\mathrm {aff}}$ on $X_{\mathbb {R}}$ is defined by

$$\begin{align*}t_\gamma w \boldsymbol{\cdot} x = w( x + \rho ) - \rho + p \gamma \end{align*}$$

for $\gamma \in \mathbb {Z}\Phi $ , $w \in W_{\mathrm {fin}}$ and $x \in X_{\mathbb {R}}$ . In the following, we recall some results about the alcove geometry associated with the dot action of $W_{\mathrm {aff}}$ on $X_{\mathbb {R}}$ ; we refer the reader to Subsections II.6.1–5 in [Reference JantzenJan03] for more details and additional references.

The affine Weyl group is generated by the affine reflections $s_{\alpha ,r} = t_{r\alpha } s_\alpha $ , for $\alpha \in \Phi ^+$ and $r \in \mathbb {Z}$ , and the fixed points of $s_{\alpha ,r}$ with respect to the p-dilated dot action form the affine hyperplane

$$\begin{align*}H_{\alpha,r} = \{ x \in X_{\mathbb{R}} \mid (x+\rho,\alpha^\vee) = pr \}. \end{align*}$$

The connected components of $X_{\mathbb {R}} \setminus \big ( \bigcup _{\alpha ,r} H_{\alpha ,r} \big )$ are called alcoves, so an alcove is any nonempty set of the form

$$\begin{align*}C = \{ x \in X_{\mathbb{R}} \mid p n_\alpha < (x+\rho,\alpha^\vee) < p \cdot (n_\alpha + 1) \text{ for all } \alpha \in \Phi^+ \} , \end{align*}$$

for some collection of integers $n_\alpha \in \mathbb {Z}$ with $\alpha \in \Phi ^+$ . The upper closure of the alcove C is defined by

$$\begin{align*}\widehat C = \{ x \in X_{\mathbb{R}} \mid p n_\alpha < (x+\rho,\alpha^\vee) \leq p \cdot (n_\alpha + 1) \text{ for all } \alpha \in \Phi^+ \}. \end{align*}$$

For every element $x \in X_{\mathbb {R}}$ , there is a unique alcove C with $x \in \widehat {C}$ . We call

$$ \begin{align*} C_0 & = \{ x \in X_{\mathbb{R}} \mid 0 < (x+\rho,\alpha^\vee) < p \text{ for all } \alpha \in \Phi^+ \} \\ & = \{ x \in X_{\mathbb{R}} \mid 0 < (x+\rho,\alpha^\vee) \text{ for all } \alpha \in \Pi \text{ and } (x+\rho,\alpha_{\mathrm{hs}}^\vee) < p \} \end{align*} $$

the fundamental alcove. The affine Weyl group acts simply transitively on the set of alcoves and the closure of every alcove is a fundamental domain. In particular, $W_{\mathrm {aff}}$ is in bijection with the set of alcoves via $w \mapsto w \boldsymbol {\cdot } C_0$ . A weight $\lambda \in X$ is called p-regular if it belongs to an alcove and p-singular if it belongs to one of the reflection hyperplanes $H_{\alpha ,r}$ with $\alpha \in \Phi ^+$ and $r \in \mathbb {Z}$ .

The affine Weyl group is a Coxeter group with simple reflections $S_{\mathrm {aff}} = S_{\mathrm {fin}} \sqcup \{ s_0 \}$ , where , and we write $\ell \colon W_{\mathrm {aff}} \to \mathbb {Z}_{\geq 0}$ for the length function. For all $x \in W_{\mathrm {aff}}$ , the coset $W_{\mathrm {fin}} x$ contains a unique element of minimal length, and we define

$$\begin{align*}W_{\mathrm{aff}}^+ = \{ x \in W_{\mathrm{aff}} \mid x \text{ has minimal length in } W_{\mathrm{fin}} x \}. \end{align*}$$

If $p \geq h$ then we also have $W_{\mathrm {aff}}^+ = \{ x \in W_{\mathrm {aff}} \mid x\boldsymbol {\cdot }0 \in X^+ \}$ . An alcove $C = w \boldsymbol {\cdot } C_0$ is called dominant if $w \in W_{\mathrm {aff}}^+$ . We say that a reflection hyperplane $H = H_{\alpha ,r}$ separates two points $x,y \in X_{\mathbb {R}}$ if either

$$\begin{align*}(x+\rho,\alpha^\vee) < pr < (y+\rho,\alpha^\vee) \qquad \text{or} \qquad (y+\rho,\alpha^\vee) < pr < (x+\rho,\alpha^\vee). \end{align*}$$

Similarly, H separates two alcoves $C , C^\prime \subseteq X_{\mathbb {R}}$ if there exist points $x \in C$ and $y \in C^\prime $ such that H separates x and y. (Equivalently, H separates x and y for all $x \in C$ and $y \in C^\prime $ .) The alcoves C and $C^\prime $ are called adjacent if there is a unique reflection hyperplane $H = H_{\alpha ,r}$ separating C and $C^\prime $ , and in that case, we have $C^\prime = s_{\alpha ,r} \boldsymbol {\cdot } C$ and $H_{\alpha ,r}$ is called a wall of C and $C^\prime $ . For every wall $H = H_{\alpha ,r}$ of C, we have either $(x+\rho ,\alpha ^\vee ) < pr$ for all $x \in C$ or $(x+\rho ,\alpha ^\vee )> pr$ for all $x \in C$ , and accordingly, we say that H belongs to the upper closure or to the lower closure of C, respectively. For an alcove $C \subseteq X_{\mathbb {R}}$ and a reflection $s = s_{\alpha ,r}$ such that $(x+\rho ,\alpha ^\vee ) < pr$ for all $x \in C$ , we write $C \uparrow s \boldsymbol {\cdot } C$ , and we define the linkage partial order $\uparrow $ on the set of alcoves as the reflexive and transitive closure of this relation; see Subsection II.6.5 in [Reference JantzenJan03].

1.4 Linkage principle and translation functors

The linkage principle asserts that two simple G-modules $L(\lambda )$ and $L(\mu )$ belong to the same block of $\mathrm {Rep}(G)$ only if the highest weights $\lambda ,\mu \in X^+$ belong to the same $W_{\mathrm {aff}}$ -orbit with respect to the p-dilated dot action; see Subsections II.7.1–3 in [Reference JantzenJan03]. For $\lambda \in \overline {C}_0 \cap X$ , we let $\mathrm {Rep}_\lambda (G)$ be the full subcategory of $\mathrm {Rep}(G)$ whose objects are the G-modules all of whose composition factors have highest weight in $W_{\mathrm {aff}} \boldsymbol {\cdot } \lambda $ (i.e., the Serre subcategory generated by the simple G-modules $L(w \boldsymbol {\cdot } \lambda )$ , for $w \in W_{\mathrm {aff}}$ such that $w\boldsymbol {\cdot }\lambda \in X^+$ ), and we call $\mathrm {Rep}_\lambda (G)$ the linkage class of $\lambda $ . Then by the linkage principle, there is a canonical projection functor

$$\begin{align*}\mathrm{pr}_\lambda \colon \mathrm{Rep}(G) \longrightarrow \mathrm{Rep}_\lambda(G) \end{align*}$$

which sends a G-module M to its largest submodule $\mathrm {pr}_\lambda M$ that belongs to $\mathrm {Rep}_\lambda (G)$ , and there is a direct sum decomposition $M = \bigoplus _{\lambda \in \overline {C}_0 \cap X} \mathrm {pr}_\lambda M$ . If the weight $\lambda \in \overline {C}_0 \cap X$ is dominant then

$$\begin{align*}L(\lambda) \cong \Delta(\lambda) \cong \nabla(\lambda) \cong T(\lambda); \end{align*}$$

see Corollary II.5.6 and Section II.E.1 in [Reference JantzenJan03].

For $\lambda ,\mu \in \overline {C}_0 \cap X$ , let $\nu \in X^+$ be the unique dominant weight in the $W_{\mathrm {fin}}$ -orbit of $\mu -\lambda $ , and consider the translation functor

$$\begin{align*}T_\lambda^\mu = \mathrm{pr}_\mu\big( L(\nu) \otimes - \big) \colon \quad \mathrm{Rep}_\lambda(G) \longrightarrow \mathrm{Rep}_\mu(G). \end{align*}$$

In the definition of $T_\lambda ^\mu $ , we can replace the simple G-module $L(\nu )$ by $\nabla (\nu )$ , $\Delta (\nu )$ or $T(\nu )$ without changing the functor, up to a natural isomorphism (see Remark 1 in Subsection II.7.6 in [Reference JantzenJan03]), and in particular, translation functors preserve the class of G-modules with Weyl filtrations and the class of tilting modules. For p-regular weights $\lambda ,\mu \in C_0$ , the translation functor $T_\lambda ^\mu $ is an equivalence with quasi-inverse $T_\mu ^\lambda $ by [Reference JantzenJan03, Proposition II.7.9], and for $w \in W_{\mathrm {aff}}^+$ , we have

$$\begin{align*}T_\lambda^\mu L(w\boldsymbol{\cdot}\lambda) \cong L(w\boldsymbol{\cdot}\mu) , \qquad T_\lambda^\mu \Delta(w\boldsymbol{\cdot}\lambda) \cong \Delta(w\boldsymbol{\cdot}\mu) , \qquad T_\lambda^\mu T(w\boldsymbol{\cdot}\lambda) \cong T(w\boldsymbol{\cdot}\mu). \end{align*}$$

Furthermore, if $\lambda \in C_0$ and $\mu \in \overline {C}_0$ then we have

(1.1) $$ \begin{align} T_\lambda^\mu \Delta(w\boldsymbol{\cdot}\lambda) \cong \begin{cases} \Delta(w\boldsymbol{\cdot}\mu) & \text{if } w\boldsymbol{\cdot}\mu \in X^+ , \\ 0 & \text{otherwise} , \end{cases} \qquad T_\lambda^\mu L(w\boldsymbol{\cdot}\lambda) \cong \begin{cases} L(w\boldsymbol{\cdot}\mu) & \text{if } w\boldsymbol{\cdot}\mu \in \widehat{ w \boldsymbol{\cdot} C_0 } , \\ 0 & \text{otherwise} , \end{cases} \end{align} $$

by Subsections II.7.11 and II.7.15 in [Reference JantzenJan03].

3.1 Reduction to p-restricted weights

Recall that we write

$$\begin{align*}X_1 = \{ \lambda \in X^+ \mid (\lambda,\alpha^\vee) < p \text{ for all } \alpha \in \Pi \} \end{align*}$$

for the set of p-restricted weights in $X^+$ . Since we assume that G is simply connected, every dominant weight $\lambda \in X^+$ can be written uniquely as $\lambda = \lambda _0 + p \lambda _1 + \cdots + p^m \lambda _m$ with $\lambda _1,\ldots ,\lambda _m \in X_1$ , and Steinberg’s tensor product theorem states that the simple G-module $L(\lambda )$ admits a tensor product decomposition

$$\begin{align*}L(\lambda) \cong L(\lambda_0) \otimes L(\lambda_1)^{[1]} \otimes \cdots \otimes L(\lambda_m)^{[m]} , \end{align*}$$

where $M \mapsto M^{[r]}$ denotes the r-th Frobenius twist functor on $\mathrm {Rep}(G)$ . See Subsections II.3.16–17 in [Reference JantzenJan03] for more details. Using Steinberg’s tensor product theorem, the problem of classifying pairs of simple modules whose tensor product is completely reducible can be reduced to pairs of simple modules with p-restricted highest weights; see Corollary 9.3 in [Reference GruberGru21].

In order to obtain an analogous reduction to p-restricted weights for classifying pairs of simple modules whose tensor product is multiplicity free, we will use the following result from Lemma 4.12 in [Reference GruberGru21], which relates multiplicity freeness and complete reducibility.

Proof. First observe that by Steinberg’s tensor product theorem, we have

$$\begin{align*}L(\lambda) \otimes L(\mu) \cong \big( L(\lambda_0) \otimes L(\mu_0) \big) \otimes \big( L(\lambda_1) \otimes L(\mu_1) \big)^{[1]} \otimes \cdots \otimes \big( L(\lambda_m) \otimes L(\mu_m) \big)^{[m]}. \end{align*}$$

Thus, if $L(\lambda _i) \otimes L(\mu _i)$ is not multiplicity free for some $i \in \{0,\ldots ,m\}$ then neither is $L(\lambda ) \otimes L(\mu )$ . Now suppose that $L(\lambda _i) \otimes L(\mu _i)$ is multiplicity free for $i=0,\ldots ,m$ . Then $L(\lambda _i) \otimes L(\mu _i)$ is completely reducible by Lemma 3.2, and furthermore, all composition factors of $L(\lambda _i) \otimes L(\mu _i)$ have p-restricted highest weights by Theorem A in [Reference GruberGru21], for $i=0,\ldots ,m$ . Let us write

$$\begin{align*}L(\lambda_i) \otimes L(\mu_i) = L(\nu_{i,1}) \oplus L(\nu_{i,2}) \oplus \cdots \oplus L(\nu_{i,m_i}) , \end{align*}$$

where the weights $\nu _{i,1},\nu _{i,2},\ldots ,\nu _{i,m_i} \in X_1$ are p-restricted and pairwise distinct. Then we have

$$ \begin{align*} L(\lambda) \otimes L(\mu) & \cong \big( L(\lambda_0) \otimes L(\mu_0) \big) \otimes \big( L(\lambda_1) \otimes L(\mu_1) \big)^{[1]} \otimes \cdots \otimes \big( L(\lambda_m) \otimes L(\mu_m) \big)^{[m]} \\ & \cong \bigotimes_{i=0}^m \big( L(\nu_{i,1}) \oplus L(\nu_{i,2}) \oplus \cdots \oplus L(\nu_{i,m_i}) \big)^{[i]} \\ & \cong \bigoplus_{\mathbf{j} = (j_0,\ldots,j_m)} L(\nu_{0,j_0}) \otimes L(\nu_{1,j_1})^{[1]} \otimes \cdots \otimes L(\nu_{m,j_m})^{[m]} \\ & \cong \bigoplus_{\mathbf{j} = (j_0,\ldots,j_m)} L(\nu_{0,j_0} + p\nu_{1,j_1} + \cdots + p^m \nu_{m,j_m}) , \end{align*} $$

where $\mathbf {j} = (j_0,\ldots ,j_m)$ runs over the tuples with $1 \leq j_i \leq m_i$ for $i=0,\ldots ,m$ . Now it is straightforward to see that the highest weights $\nu _{\mathbf {j}} = \nu _{0,j_0} + p\nu _{1,j_1} + \cdots + p^m \nu _{m,j_m}$ of the composition factors of $L(\lambda ) \otimes L(\mu )$ are pairwise distinct, and so $L(\lambda ) \otimes L(\mu )$ is multiplicity free, as required.

3.2 Good filtration dimension

Following [Reference Friedlander and ParshallFP86, Section 3], we define the good filtration dimension of a nonzero G-module M as

$$\begin{align*}\mathrm{gfd}\, M = \max\big\{ d>0 \mathrel{\big|} \mathrm{Ext}_G^d( \Delta(\lambda) , M ) \neq 0 \text{ for some } \lambda \in X^+ \big\}. \end{align*}$$

Note that $\mathrm {gfd}\, M = 0$ if and only if M admits a good filtration, by Proposition II.4.16 in [Reference JantzenJan03]. The following lemma shows that the good filtration dimension is well-behaved with respect to tensor products and direct sums of G-modules.

Lemma 3.4. For G-modules M and N, we have

$$\begin{align*}\mathrm{gfd}(M \otimes N) \leq \mathrm{gfd}(M) + \mathrm{gfd}(N) \qquad \text{and} \qquad \mathrm{gfd}(M \oplus N) = \max\{ \mathrm{gfd}(M) , \mathrm{gfd}(N) \}. \end{align*}$$

The good filtration dimension of simple G-modules with p-regular highest weights was determined by A. Parker in [Reference ParkerPar03, Corollary 4.5].

Using the two preceding lemmas, we obtain a new necessary condition for the complete reducibility of tensor products of simple G-modules with p-regular highest weights.

Proof. If $L(x\boldsymbol {\cdot }\lambda ) \otimes L(y\boldsymbol {\cdot }\mu )$ is completely reducible then $L(z\boldsymbol {\cdot }\nu )$ is a direct summand of $L(x\boldsymbol {\cdot }\lambda ) \otimes L(y\boldsymbol {\cdot }\mu )$ , and using Lemmas 3.4 and 3.5, we obtain

$$\begin{align*}\ell(z) = \mathrm{gfd} \, L(z\boldsymbol{\cdot}\nu) \leq \mathrm{gfd}\big( L(x\boldsymbol{\cdot}\lambda) \otimes L(y\boldsymbol{\cdot}\mu) \big) \leq \mathrm{gfd} \, L(x\boldsymbol{\cdot}\lambda) + \mathrm{gfd} \, L(y\boldsymbol{\cdot}\mu) = \ell(x) + \ell(y) , \end{align*}$$

as claimed.

3.3 Singular and regular modules

A tilting module $T = \bigoplus _{\mu \in X^+} T(\mu )^{\oplus m_\mu }$ is called negligible if we have $m_\mu = 0$ for all $\mu \in C_0$ . Following Section 2 in [Reference GruberGru24], we call a G-module M singular if there exists a bounded complex

$$\begin{align*}C = ( \cdots \to T_{-1} \to T_0 \to T_1 \to \cdots ) \end{align*}$$

of negligible tilting modules such that $H^0(C) \cong M$ and $H^i(C) = 0$ for $i \neq 0$ , and we say that M is regular if M is not singular. The singular G-modules form a thick tensor ideal in $\mathrm {Rep}(G)$ , that is, for G-modules M and N, we have

1. (1) if M is singular then $M \otimes N$ is singular;

2. (2) $M \oplus N$ is singular if and only if both M and N are singular.

Furthermore, for $\lambda \in X^+$ , the simple G-module $L(\lambda )$ is singular if and only if the weight $\lambda $ is p-singular, see Lemma 3.3 in [Reference GruberGru24]. This observation gives rise to the following necessary condition for complete reducibility of tensor products of simple G-modules.

For every G-module M, we can write

$$\begin{align*}M = M_{\mathrm{sing}} \oplus M_{\mathrm{reg}} , \end{align*}$$

where $M_{\mathrm {sing}}$ is the direct sum of all singular indecomposable direct summands of M and $M_{\mathrm {reg}}$ is the direct sum of all regular indecomposable direct summands of M, for a fixed Krull–Schmidt decomposition of M. This direct sum decomposition into a singular part and a regular part has been used in [Reference GruberGru24] to describe how tensor products of G-modules interact with the decomposition of G into linkage classes and with translation functors, on the level of regular parts. In order to state these results, we define the Verlinde coefficient $c_{\lambda ,\mu }^\nu $ , for $\lambda ,\mu ,\nu \in C_0 \cap X$ , as the multiplicity

$$\begin{align*}c_{\lambda,\mu}^\nu = \big[ T(\lambda) \otimes T(\mu) : T(\nu) \big]_\oplus \end{align*}$$

of $T(\nu )$ in a Krull–Schmidt decomposition of $T(\lambda ) \otimes T(\mu )$ . We have the following linkage principle and translation principle for tensor products; see Lemma 3.12 and Theorem 3.14 in [Reference GruberGru24].

The coefficients $c_{\lambda ,\mu }^\nu $ are the structure coefficients of the so called Verlinde category $\mathrm {Ver}(G)$ of G, that is, the quotient of the subcategory $\mathrm {Tilt}(G)$ of tilting G-modules by the thick tensor ideal of negligible tilting modules [Reference Georgiev and MathieuGM94, Reference Andersen and ParadowskiAP95, Reference Etingof and OstrikEO22]. The category $\mathrm {Ver}(G)$ is semisimple, with simple objects $T(\delta )$ for $\delta \in C_0 \cap X$ . We next recall two elementary results that will be useful for computing Verlinde coefficients in concrete examples. Let us write $W_{\mathrm {ext}} = X \rtimes W_{\mathrm {fin}}$ for the extended affine Weyl group and $\Omega = \mathrm {Stab}_{W_{\mathrm {ext}}}(C_0)$ , so that

$$\begin{align*}\Omega \cong W_{\mathrm{ext}} / W_{\mathrm{aff}} \cong X / \mathbb{Z}\Phi. \end{align*}$$

The Verlinde coefficients are invariant under the action of $\Omega $ on $C_0$ by Lemma 1.1 in [Reference GruberGru24].

Lemma 3.9. For $\lambda ,\mu ,\nu \in C_0 \cap X$ and $\omega \in \Omega $ , we have

$$\begin{align*}c_{\omega \boldsymbol{\cdot} \lambda , \mu}^{\omega\boldsymbol{\cdot}\nu} = c_{\lambda , \omega \boldsymbol{\cdot} \mu}^{\omega\boldsymbol{\cdot}\nu} = c_{\lambda,\mu}^\nu. \end{align*}$$

Recall that we write $w_0$ for the longest element in $W_{\mathrm {fin}}$ . The Verlinde coefficients also have the following exchange property:

Proof. The dual of $T(\mu )$ is $T(\mu )^* \cong T(-w_0(\mu ))$ , and as the Verlinde category is semisimple with simple objects $T(\delta )$ for $\delta \in C_0 \cap X$ , we have

$$\begin{align*}c_{\lambda,\mu}^\nu = \dim \mathrm{Hom}_{\mathrm{Ver}(G)}\big( T(\lambda) \otimes T(\mu) , T(\nu) \big) = \dim \mathrm{Hom}_{\mathrm{Ver}(G)}\big( T(\lambda) , T( - w_0(\mu) ) \otimes T(\nu) \big) = c_{\nu,-w_0(\mu)}^\lambda , \end{align*}$$

as claimed.

The following lemma relates the Verlinde coefficients to multiplicity freeness of tensor products of simple G-modules.

Lemma 3.11. For $\lambda ,\mu ,\nu \in C_0 \cap X$ and $x \in W_{\mathrm {aff}}^+$ , the multiplicity of $L(x\boldsymbol {\cdot }\nu )$ in a Krull–Schmidt decomposition of $L(x\boldsymbol {\cdot }\lambda ) \otimes L(\mu )$ is given by

$$\begin{align*}\big[ L(x\boldsymbol{\cdot}\lambda) \otimes L(\mu) : L(x\boldsymbol{\cdot}\nu) \big]_\oplus = c_{\lambda,\mu}^\nu. \end{align*}$$

In particular, if $L(x\boldsymbol {\cdot }\lambda ) \otimes L(\mu )$ is multiplicity free then $c_{\lambda ,\mu }^\nu \leq 1$ for all $\nu \in C_0 \cap X$ .

Proof. We clearly have $\big ( L(x\boldsymbol {\cdot }0) \otimes L(0) \big )_{\mathrm {reg}} \cong L(x\boldsymbol {\cdot }0)$ , and so Theorem 3.8 yields

$$\begin{align*}\big( L(x\boldsymbol{\cdot}\lambda) \otimes L(\mu) \big)_{\mathrm{reg}} \cong \bigoplus_{\nu \in C_0 \cap X} T_0^\nu \big( L(x\boldsymbol{\cdot}0) \otimes L(0) \big)_{\mathrm{reg}}^{\oplus c_{\lambda,\mu}^\nu} \cong \bigoplus_{\nu \in C_0 \cap X} T_0^\nu L(x\boldsymbol{\cdot}0)^{\oplus c_{\lambda,\mu}^\nu} \cong \bigoplus_{\nu \in C_0 \cap X} L(x\boldsymbol{\cdot}\nu)^{\oplus c_{\lambda,\mu}^\nu}. \end{align*}$$

As $L(x\boldsymbol {\cdot }\nu )$ is regular, the multiplicity of $L(x\boldsymbol {\cdot }\nu )$ in a Krull–Schmidt decomposition of $L(x\boldsymbol {\cdot }\lambda ) \otimes L(\mu )$ coincides with the multiplicity of $L(x\boldsymbol {\cdot }\nu )$ in a Krull–Schmidt decomposition of $\big ( L(x\boldsymbol {\cdot }\lambda ) \otimes L(\mu ) \big )_{\mathrm {reg}}$ , and the claim follows.

3.4 Tensor products and weight space dimensions

In this subsection, we prove some results that relate multiplicities in tensor products to dimensions of weight spaces. This will be useful later on to establish the multiplicity freeness of certain tensor products of simple G-modules.

Lemma 3.12. Let M be a G-module and $\lambda \in X^+$ such that $L(\lambda )\otimes M$ is completely reducible. For all weights $\mu \in X^+$ , we have

$$\begin{align*}[ L(\lambda) \otimes M : L(\mu) ] \leq \dim M_{\mu-\lambda}. \end{align*}$$

Proof. Since $L(\lambda ) \otimes M$ is completely reducible, we have

$$ \begin{align*} [ L(\lambda) \otimes M : L(\mu) ] & = \dim \mathrm{Hom}_G\big( L(\mu) , L(\lambda) \otimes M \big) \\ & \leq \dim \mathrm{Hom}_G\big( \Delta(\mu) , \nabla(\lambda) \otimes M \big) \\ & \leq \dim M_{\mu-\lambda} , \end{align*} $$

where the second inequality follows from Theorem 2.5 in [Reference Brundan and KleshchevBK99].

Recall that a weight $\varpi \in X^+$ is called minuscule if all weights of $\nabla (\varpi )$ belong to the same $W_{\mathrm {fin}}$ -orbit, and that $L(\varpi ) = \nabla (\varpi )$ if $\varpi \in X^+$ is minuscule (see Section II.2.15 in [Reference JantzenJan03]). The following two results describe tensor products where one factor is simple with minuscule highest weight.

Lemma 3.14. Let $\lambda \in \overline {C}_0 \cap X$ and let $\varpi \in X^+$ be minuscule. Then for every G-module M in the linkage class $\mathrm {Rep}_\lambda (G)$ , we have

$$\begin{align*}L(\varpi) \otimes M \cong \bigoplus_{\varpi^\prime} T_\lambda^{\lambda+\varpi^\prime} M , \end{align*}$$

where $\varpi ^\prime $ runs over the weights in $W_{\mathrm {fin}} \varpi $ such that $\lambda +\varpi ^\prime \in \overline {C}_0$ .

Proof. By the linkage principle, we have

$$\begin{align*}L(\varpi) \otimes M \cong \bigoplus_{\nu \in \overline{C}_0 \cap X} \mathrm{pr}_\nu\big( L(\varpi) \otimes M \big). \end{align*}$$

For a weight $\nu \in \overline {C}_0 \cap X$ such that $\mathrm {pr}_\nu \big ( L(\varpi ) \otimes M \big ) \neq 0$ , there exists $x \in W_{\mathrm {aff}}$ such that $x \boldsymbol {\cdot }\nu -\lambda $ is a weight of $L(\varpi )$ by Lemma II.7.5 in [Reference JantzenJan03], and by Corollary 2.5, there further exists $y \in W_{\mathrm {aff}}$ such that $yx\boldsymbol {\cdot }\nu -\lambda $ is a weight of $L(\varpi )$ and $yx\boldsymbol {\cdot }\nu \in \overline {C}_0$ . This forces $yx\boldsymbol {\cdot }\nu = \nu $ because $\overline {C}_0$ is a fundamental domain for the p-dilated dot action of $W_{\mathrm {aff}}$ on $X_{\mathbb {R}}$ , and we conclude that $\mathrm {pr}_\nu \big ( L(\varpi ) \otimes M \big ) \neq 0$ only if $\nu -\lambda $ is a weight of $L(\varpi )$ . For a weight $\varpi ^\prime $ of $L(\varpi )$ such that $\lambda + \varpi ^\prime \in \overline {C}_0$ , we have $\mathrm {pr}_\nu \big ( L(\varpi ) \otimes M \big ) \cong T_\lambda ^{\lambda +\varpi ^\prime } M$ because $\varpi $ is the unique dominant weight in the $W_{\mathrm {fin}}$ -orbit of $\varpi ^\prime $ , and the claim follows.

Corollary 3.15. Let $\lambda \in X^+$ be a p-regular weight and let C be the unique alcove containing $\lambda $ . For a minuscule weight $\varpi \in X^+$ , we have

$$\begin{align*}L(\lambda) \otimes L(\varpi) \cong \bigoplus_{\varpi^\prime} L(\lambda+\varpi^\prime) , \end{align*}$$

where $\varpi ^\prime $ runs over the weights in $W_{\mathrm {fin}} \varpi $ such that $\lambda +\varpi ^\prime \in \widehat {C}$ .

Proof. Let $x \in W_{\mathrm {aff}}$ such that $x \boldsymbol {\cdot } C_0 = C$ and set $\lambda _0 = x^{-1} \boldsymbol {\cdot } \lambda \in C_0$ . By Lemma 3.14, we have

$$\begin{align*}L(\lambda) \otimes L(\varpi) = L(x\boldsymbol{\cdot}\lambda_0) \otimes L(\varpi) \cong \bigoplus_{\varpi^\prime} T_{\lambda_0}^{\lambda_0+\varpi^\prime} L(x\boldsymbol{\cdot}\lambda_0) , \end{align*}$$

where $\varpi ^\prime $ runs over the weights in $W_{\mathrm {fin}} \varpi $ such that $\lambda _0+\varpi ^\prime \in \overline {C}_0$ . Further let $w \in W_{\mathrm {fin}}$ and $\gamma \in \mathbb {Z}\Phi $ such that $x = t_\gamma w$ , and observe that $x \boldsymbol {\cdot } (\lambda _0+\varpi ^\prime ) = x\boldsymbol {\cdot }\lambda _0 + w(\varpi ^\prime ) = \lambda + w(\varpi ^\prime )$ . We have

$$\begin{align*}T_{\lambda_0}^{\lambda_0+\varpi^\prime} L(x\boldsymbol{\cdot}\lambda_0) \cong \begin{cases} L( \lambda + w(\varpi^\prime) ) & \text{if } \lambda + w(\varpi^\prime) \in \widehat{C} \\ 0 & \text{otherwise} \end{cases} \end{align*}$$

by (1.1), and the claim follows because w permutes the weight in $W_{\mathrm {fin}} \varpi $ .

In Section 5 below, we will need to explicitly compute the characters of certain tensor products of the form $\Delta (\lambda ) \otimes \Delta (\mu )$ , with $\lambda ,\mu \in X^+$ . We will use the following formula in terms of weight space dimensions; see Lemma II.5.8 in [Reference JantzenJan03].

Proposition 3.16. For every G-module M and all $\lambda \in X^+$ , we have

$$\begin{align*}\operatorname{\mathrm{ch}} M \cdot \chi(\lambda) = \sum_{\nu\in X} \dim M_\nu \cdot \chi(\lambda+\nu). \end{align*}$$

3.5 Comparison with characteristic zero

Let us write $G_{\mathbb {C}}$ for the unique simply connected simple algebraic group over the complex numbers $\mathbb {C}$ with root system $\Phi $ . In this subsection, we explain how the composition multiplicities in tensor products of simple $G_{\mathbb {C}}$ -modules can be used to study multiplicity freeness of tensor products of simple G-modules. For $\lambda \in X^+$ , denote by $L_{\mathbb {C}}(\lambda )$ the simple $G_{\mathbb {C}}$ -module of highest weight $\lambda $ , and note that the character $\operatorname {\mathrm {ch}} L_{\mathbb {C}}(\lambda ) = \chi (\lambda ) = \operatorname {\mathrm {ch}} \nabla (\lambda )$ is given by Weyl’s character formula. Therefore, we can write

$$\begin{align*}\sum_{\nu \in X^+} \big[ L_{\mathbb{C}}(\lambda) \otimes L_{\mathbb{C}}(\mu) : L_{\mathbb{C}}(\nu) \big] \cdot \chi(\nu) = \chi(\lambda) \cdot \chi(\mu) = \sum_{\nu \in X^+} \big[ \nabla(\lambda) \otimes \nabla(\mu) : \nabla(\nu) \big]_\nabla \cdot \chi(\nu) , \end{align*}$$

and it follows that

$$\begin{align*}\big[ L_{\mathbb{C}}(\lambda) \otimes L_{\mathbb{C}}(\mu) : L_{\mathbb{C}}(\nu) \big] = \big[ \nabla(\lambda) \otimes \nabla(\mu) : \nabla(\nu) \big]_\nabla = \dim \mathrm{Hom}_G\big( \Delta(\nu) , \nabla(\lambda) \otimes \nabla(\mu) \big) \end{align*}$$

for all $\lambda ,\mu ,\nu \in X^+$ . This observation is crucial for the proofs of the two following results.

Proof. For all $\nu \in C_0 \cap X$ , we have

$$ \begin{align*} c_{\lambda,\mu}^\nu = [ T(\lambda) \otimes T(\mu) : T(\nu) ]_\oplus &\leq [ T(\lambda) \otimes T(\mu) : \nabla(\nu) ]_\nabla \\ & =[ \nabla(\lambda) \otimes \nabla(\mu) : \nabla(\nu) ]_\nabla = [L_{\mathbb{C}}(\lambda)\otimes L_{\mathbb{C}}(\mu):L_{\mathbb{C}}(\nu)]. \qquad \end{align*} $$

Thus if $[L_{\mathbb {C}}(\lambda )\otimes L_{\mathbb {C}}(\mu ):L_{\mathbb {C}}(\nu )]\leq 1$ for all $\nu \in C_0 \cap X$ , we conclude that $c_{\lambda ,\mu }^\nu \leq 1$ for all $\nu \in C_0 \cap X$ .

Proof. If $L(\lambda ) \otimes L(\mu )$ is completely reducible then we have

$$ \begin{align*} [ L(\lambda) \otimes L(\mu) : L(\nu) ] & = \dim \mathrm{Hom}_G\big( L(\nu) , L(\lambda) \otimes L(\mu) \big) \\ & \quad \leq \dim \mathrm{Hom}_G\big( \Delta(\nu) , \nabla(\lambda) \otimes \nabla(\mu) \big) = [ L_{\mathbb{C}}(\lambda) \otimes L_{\mathbb{C}}(\mu) : L_{\mathbb{C}}(\nu) ] \end{align*} $$

for all $\nu \in X^+$ , and the claim is immediate.

The comparison with multiplicities in characteristic zero also allows us to observe the following monotonicity property for good filtration multiplicities in tensor products of induced modules and Weyl filtration multiplicities in tensor products of Weyl modules.

Lemma 3.19. For $\lambda ,\mu ,\nu ,\delta \in X^+$ , we have

$$ \begin{align*} \big[ \nabla(\lambda+\delta) \otimes \nabla(\mu) : \nabla(\nu+\delta) \big]_\nabla & \geq \big[ \nabla(\lambda) \otimes \nabla(\mu) : \nabla(\nu) \big]_\nabla , \\[3pt] \big[ \Delta(\lambda+\delta) \otimes \Delta(\mu) : \Delta(\nu+\delta) \big]_\Delta & \geq \big[ \Delta(\lambda) \otimes \Delta(\mu) : \Delta(\nu) \big]_\Delta. \end{align*} $$

Proof. As observed at the beginning of the subsection, we have

$$\begin{align*}\big[ L_{\mathbb{C}}(\lambda) \otimes L_{\mathbb{C}}(\mu) : L_{\mathbb{C}}(\nu) \big] = \big[ \nabla(\lambda) \otimes \nabla(\mu) : \nabla(\nu) \big]_\nabla \end{align*}$$

for all $\lambda ,\mu ,\nu \in X^+$ , and the first inequality follows from Proposition 2.9 in [Reference StembridgeSte03]. The second inequality follows by taking duals.

4.1 Setup

The positive roots for $G = \mathrm {SL}_3(\Bbbk )$ can be written as $\Phi ^+ = \{ \alpha _1 , \alpha _2 , \alpha _{\mathrm {h}} \}$ , where $\alpha _1$ and $\alpha _2$ are the simple roots and $\alpha _{\mathrm {h}} = \alpha _1 + \alpha _2$ is the highest root. The weight lattice X is spanned by the fundamental dominant weights $\varpi _1$ and $\varpi _2$ . The weights $\varpi _1$ and $\varpi _2$ are minuscule, and we have

$$\begin{align*}\alpha_1 = 2 \varpi_1 - \varpi_2 , \qquad \alpha_2 = - \varpi_1 + 2 \varpi_2 , \qquad \alpha_{\mathrm{h}} = \varpi_1 + \varpi_2 = \rho. \end{align*}$$

We write $S_{\mathrm {fin}} = \{t,u\}$ for the set of simple reflections in $W_{\mathrm {fin}}$ , with $t = s_{\alpha _1}$ and $u = s_{\alpha _2}$ . The affine simple reflection is denoted by $s = s_0 = t_{\alpha _{\mathrm {h}}} s_{\alpha _{\mathrm {h}}}$ , so that $S_{\mathrm {aff}} = \{ s,t,u \}$ is the set of simple reflections in $W_{\mathrm {aff}}$ . We fix labelings for certain alcoves as in the figure below, so that

$$\begin{align*}C_1 = s \boldsymbol{\cdot} C_0 , \qquad C_{2a} = su \boldsymbol{\cdot} C_0 , \qquad C_{2b} = st \boldsymbol{\cdot} C_0. \end{align*}$$

Alternatively, the labeled alcoves can be described as follows:

$$ \begin{align*} C_0 & = \{ a \varpi_1 + b \varpi_2 \mid a> -1 , ~ b > -1 , ~ a+b < p-2 \} , \\ C_1 & = \{ a \varpi_1 + b \varpi_2 \mid a+b> p-2 , ~ a < p-1 , ~ b < p-1 \} , \\ C_{2a} & = \{ a \varpi_1 + b \varpi_2 \mid a> p-1 , ~ b > -1 , ~ a+b < 2p-2 \} , \\ C_{2b} & = \{ a \varpi_1 + b \varpi_2 \mid a> -1 , ~ b > p-1 , ~ a+b < 2p-2 \}. \end{align*} $$

Observe that all p-restricted weights belong to the upper closure of one of the alcoves $C_0$ and $C_1$ . The wall separating two alcoves $C_x$ and $C_y$ will be denoted by $F_{x,y} = F_{y,x}$ , for $x,y \in \{0,1,2a,2b\}$ . In the following, we give a concrete example of what the reflection smallness condition from Section 2 means for a weight in one of the alcoves $C_0$ and $C_1$ .

Example 4.1. Let $\lambda ,\mu \in X^+$ and write $\lambda = a \varpi _1 + b \varpi _2$ and $\mu = a^\prime \varpi _1 + b^\prime \varpi _2$ . Then we have

$$\begin{align*}W_{\mathrm{fin}} \mu = \big\{ c \varpi_1 + d \varpi_2 \mathrel{\big|} \pm (c,d) \in \{ (a^\prime,b^\prime) , (a^\prime+b^\prime,-b^\prime) , (-a^\prime,a^\prime+b^\prime) \} \big \}. \end{align*}$$

The alcove $C_0$ has a unique wall $F_{0,1} = H_{\alpha _{\mathrm {h}},1}$ that belongs to its upper closure, and so if $\lambda \in \widehat {C}_0$ then $\mu $ is reflection small with respect to $\lambda $ if and only if $\lambda +w\mu \leq s_{\alpha _{\mathrm {h}},1} \boldsymbol {\cdot } (\lambda +w\mu )$ for all $w \in W_{\mathrm {fin}}$ , or equivalently, $(\lambda +w\mu +\rho ,\alpha _{\mathrm {h}}^\vee ) \leq p$ for all $w \in W_{\mathrm {fin}}$ . As $(\lambda +\rho ,\alpha _{\mathrm {h}}^\vee ) = a+b+2$ and $(w\mu ,\alpha _{\mathrm {h}}^\vee ) \in \{ \pm (a'+b') , \pm a' , \pm b' \}$ , we conclude that $\mu $ is reflection small with respect to $\lambda $ if and only if

$$\begin{align*}a + b + a^\prime + b^\prime \leq p-2. \end{align*}$$

Similarly, the walls that belong to the upper closure of $C_1$ are precisely $F_{1,2a} = H_{\alpha _1,1}$ and $F_{1,2b} = H_{\alpha _2,1}$ , and so if $\lambda \in \widehat {C}_1$ then $\mu $ is reflection small with respect to $\lambda $ if and only if $(\lambda +w\mu +\rho ,\alpha _1^\vee ) \leq p$ and $(\lambda +w\mu +\rho ,\alpha _2^\vee ) \leq p$ for all $w \in W_{\mathrm {fin}}$ . This is easily seen to be equivalent to the conditions

$$\begin{align*}a+a^\prime+b^\prime \leq p-1 \qquad \text{and} \qquad b+a^\prime+b^\prime \leq p-1. \end{align*}$$

4.2 Main results

We are now ready to state our main results about complete reducibility and multiplicity freeness of tensor products of simple modules for $G = \mathrm {SL}_3(\Bbbk )$ . The proofs will be given at the end of Subsections 4.4 and 4.5, respectively. Recall from Subsection 3.1 that it suffices to consider pairs of simple G-modules with p-restricted highest weights.

Theorem 4.2. For $\lambda , \mu \in X_1 \setminus \{0\}$ , the tensor product $L(\lambda ) \otimes L(\mu )$ is completely reducible if and only if $\lambda $ and $\mu $ satisfy one of the conditions in Table 2, up to interchanging $\lambda $ and $\mu $ .

Table 2 Weights $\lambda ,\mu \in X_1 \setminus \{ 0 \}$ such that $L(\lambda ) \otimes L(\mu )$ is completely reducible, for G of type $\mathrm {A}_2$ .

Theorem 4.3. For $\lambda , \mu \in X_1 \setminus \{0\}$ , the tensor product $L(\lambda ) \otimes L(\mu )$ is multiplicity free if and only if $\lambda $ and $\mu $ satisfy one of the conditions in Table 3, up to interchanging $\lambda $ and $\mu $ .

Table 3 Weights $\lambda ,\mu \in X_1 \setminus \{ 0 \}$ such that $L(\lambda ) \otimes L(\mu )$ is multiplicity free, for G of type $\mathrm {A}_2$ .

4.3 Structure of Weyl modules and tilting modules

In order to prove our main results, we will need some information about the submodule structure of certain Weyl modules and indecomposable tilting modules for $G = \mathrm {SL}_3(\Bbbk )$ . The structure of these modules has been computed by Bowman, Doty, and Martin in [Reference Bowman, Doty and MartinBDM11, Reference Bowman, Doty and MartinBDM15]. We recall some of their results below.

Proposition 4.4. For $\lambda \in C_0 \cap X$ , the Weyl modules $\Delta (w \boldsymbol {\cdot } \lambda )$ with $w \in \{ e , s , st , su \}$ are all uniserialFootnote ¹ , with respective composition series given by $\Delta (\lambda ) = [ L(\lambda ) ]$ and

$$\begin{align*}\Delta(s\boldsymbol{\cdot}\lambda) = [ L(s\boldsymbol{\cdot}\lambda) , L(\lambda) ] , \qquad\!\! \Delta(st \boldsymbol{\cdot} \lambda) = [ L(st \boldsymbol{\cdot}\lambda) , L(s \boldsymbol{\cdot}\lambda) ] , \qquad\!\! \Delta(su\boldsymbol{\cdot}\lambda) = [ L(su \boldsymbol{\cdot}\lambda) , L(s \boldsymbol{\cdot}\lambda) ]. \end{align*}$$

The Weyl modules with p-singular p-restricted highest weights are all simple.

Proposition 4.5. For $\lambda \in C_0 \cap X$ , we have $T(\lambda ) = \Delta (\lambda ) = L(\lambda )$ and $T(s\boldsymbol {\cdot }\lambda )$ is uniserial with composition series $T(s\boldsymbol {\cdot }\lambda ) = [ L(\lambda ) , L(s\boldsymbol {\cdot }\lambda ) , L(\lambda ) ]$ . The structure of $T(st\boldsymbol {\cdot }\lambda )$ and $T(su\boldsymbol {\cdot }\lambda )$ is displayed in the following diagrams, where we replace a simple G-module $L(w\boldsymbol {\cdot }\lambda )$ by the label $w \in W_{\mathrm {aff}}^+$ :

The indecomposable tilting modules with p-singular p-restricted highest weights are all simple.

4.4 Proofs: complete reducibility

In this subsection, we establish all the necessary preliminary results for proving Theorem 4.2. The proof of the theorem is given at the end of the subsection (see page 25).

Remark 4.6. Let $\lambda ,\mu \in X_1$ and write $\lambda = a\varpi _1 + b\varpi _2$ and $\mu = a^\prime \varpi _1 + b^\prime \varpi _2$ , with $0 \leq a,b,a^\prime ,b^\prime < p$ . Suppose that $L(\lambda ) \otimes L(\mu )$ is completely reducible. Then all composition factors of $L(\lambda ) \otimes L(\mu )$ have p-restricted highest weights by Theorem A in [Reference GruberGru21]. By truncation to the two Levi subgroups of type $\mathrm {A}_1$ corresponding to the simple roots $\alpha _1$ and $\alpha _2$ respectively, we see that $L(\lambda +\mu - c\alpha _1)$ and $L(\lambda +\mu -d\alpha _2)$ are composition factors of $L(\lambda ) \otimes L(\mu )$ for all $0 \leq c \leq \min \{b,b^\prime \}$ and $0 \leq d \leq \min \{a,a^\prime \}$ , each appearing with composition multiplicity one (cf. Remark 4.13 in [Reference GruberGru21]). In particular, the weights $\lambda +\mu - c \alpha _1$ and $\lambda +\mu - d \alpha _2$ are p-restricted, and it follows that

$$ \begin{align*} a + a^\prime + \min\{ b , b^\prime \} & \leq p-1 , \\ b + b^\prime + \min\{ a , a^\prime \} & \leq p-1. \end{align*} $$

We next compute two explicit direct sum decompositions of tensor products. These will be used in Lemma 4.8 below to establish the complete reducibility of tensor products of simple G-modules that satisfy the condition 1 in Table 2

Proof. By Subsection 4.3, we have $L( (p-1) \cdot \varpi _2 ) \cong \Delta ( (p-1) \cdot \varpi _2 )$ , and using Lemma 3.14 and Proposition II.7.13 in [Reference JantzenJan03], it follows that

$$\begin{align*}L( (p-1) \cdot \varpi_2 ) \otimes L(\varpi_1) \cong \Delta( (p-1) \cdot \varpi_1 + \varpi_2 ) \oplus \Delta( (p-2) \cdot \varpi_2 ). \end{align*}$$

Now the first claim follows because

$$\begin{align*}\Delta( (p-1) \cdot \varpi_1 + \varpi_2 ) = L( (p-1) \cdot \varpi_1 + \varpi_2 ) , \qquad \Delta( (p-2) \cdot \varpi_2 ) = L( (p-2) \cdot \varpi_2 ) , \end{align*}$$

again by Subsection 4.3. The proof of (2) is analogous.

Our next goal is to prove that a tensor product $L(\lambda ) \otimes L(\mu )$ with $\lambda \in \widehat {C}_1$ and $\mu \in X_1$ is completely reducible if and only if one of the conditions 2–4 or 2*–3* from Table 2 are satisfied. This will follow from Lemma 4.9 and Proposition 4.11 below.

Proof. If $a^\prime \leq a$ and $b^\prime \leq b$ then $a + a^\prime + b^\prime \leq p-1$ and $b + a^\prime + b^\prime \leq p-1$ by Remark 4.6, and so $\mu $ is reflection small with respect to $\lambda $ (see Example 4.1). If $b^\prime>b$ then Remark 4.6 and the fact that $\lambda $ belongs to the upper closure of $C_1$ imply that

$$\begin{align*}p-1 \leq a+b \leq a+b+a^\prime \leq p-1 , \end{align*}$$

and so $a^\prime = 0$ and $a+b = p-1$ . The inequality $b^\prime \leq p-1-b = a$ also follows from Remark 4.6, and the case $a^\prime>a$ is analogous.

Proof. First suppose that $b^\prime> b+1$ and consider the weight $\nu = (p-1-b^\prime +b) \cdot \varpi _1 \in C_0$ . By the results in Subsection 4.3, we have $L(\mu ) = T(\mu )$ and $L(\nu ) = T(\nu )$ , and it is straightforward to see using weight considerations that the indecomposable tilting module of highest weight

$$\begin{align*}-w_0(\mu)+\nu = (p-1+b) \cdot \varpi_1 = sus \boldsymbol{\cdot} \lambda \in C_{2a} \end{align*}$$

is a direct summand of $L(\mu )^* \otimes L(\nu )$ . Since $L(\lambda )$ is a submodule of $T(sus\boldsymbol {\cdot }\lambda )$ by Subsection 4.3, we conclude that

$$\begin{align*}0 \neq \mathrm{Hom}_G\big( L(\lambda) , L(\mu)^* \otimes L(\nu) \big) \cong \mathrm{Hom}_G\big( L(\lambda) \otimes L(\mu) , L(\nu) \big). \end{align*}$$

The non-negligible tilting module $T(\nu ) \cong L(\nu )$ cannot be a direct summand of $L(\lambda ) \otimes L(\mu )$ by Proposition 4.10, and we conclude that $L(\lambda ) \otimes L(\mu )$ is not completely reducible.

It remains to show that $L(\lambda ) \otimes L(\mu )$ is completely reducible if $b^\prime =b+1$ . To that end, observe that $L(\mu ) = L( (b+1) \cdot \varpi _2 )$ is a direct summand of $L(b\varpi _2) \otimes L(\varpi _2)$ , so every indecomposable direct summand M of $L(\lambda ) \otimes L(\mu )$ is also a direct summand of $L(\lambda ) \otimes L(b\varpi _2) \otimes L(\varpi _2)$ . Now $b \varpi _2$ is reflection small with respect to $\lambda $ , so $L(\lambda ) \otimes L(b\varpi _2)$ is a direct sum of simple G-modules with highest weights in $\widehat {C}_1$ by Theorem 2.14, and by weight considerations, it is straightforward to see that the only simple direct summand of $L(\lambda ) \otimes L(b\varpi _2)$ with highest weight in $\widehat {C}_1 \setminus C_1$ is $L((p-1)\cdot \varpi _1)$ . We conclude that M is a direct summand of $L(\nu ) \otimes L(\varpi _1)$ for some weight $\nu \in C_1 \cup \{ (p-1) \cdot \varpi _1 \}$ . Now $L(\nu ) \otimes L(\varpi _1)$ is completely reducible by Corollary 3.15 (for $\nu \in C_1$ ) and Lemma 4.7 (for $\nu = (p-1) \cdot \varpi _2$ ), so M is simple and $L(\lambda ) \otimes L(\mu )$ is completely reducible, as required.

Before we can give the proof of Theorem 4.2, it remains to consider tensor products of simple G-modules with highest weights in the upper closure of $C_0$ .

Proof. As $\lambda ,\mu \in \widehat {C}_0$ , we have $L(\lambda ) \cong T(\lambda )$ and $L(\mu ) \cong T(\mu )$ , and it follows that $T(\lambda +\mu )$ is a direct summand of $L(\lambda ) \otimes L(\mu ) \cong T(\lambda ) \otimes T(\mu )$ . As the indecomposable tilting modules with highest weights in $C_1$ are nonsimple (see Subsection 4.3) and as all composition factors of $L(\lambda ) \otimes L(\mu )$ have p-restricted highest weights (see Remark 4.6), this implies that either $\lambda +\mu $ belongs to the upper closure of $C_0$ or $\lambda +\mu $ belongs to one of the walls $F_{1,2a}$ and $F_{1,2b}$ of $C_1$ . Observe that the first condition is equivalent to $\mu $ being reflection small with respect to $\lambda $ by Example 4.1.

Now assume without loss of generality that $\lambda + \mu \in F_{1,2a}$ (the case $\lambda +\mu \in F_{1,2b}$ being analogous). If we write $\lambda = a \varpi _1 + b \varpi _2$ and $\mu = a^\prime \varpi _1 + b^\prime \varpi _2$ then this means that $a+a^\prime = p-1$ . Since we assume that $\lambda $ and $\mu $ belong to the upper closure of $C_0$ , we further have $a+b \leq p-2$ and $a^\prime +b^\prime \leq p-2$ , and it follows that $a,a^\prime \neq 0$ and

$$\begin{align*}b + b^\prime \leq 2p-4 - (a+a^\prime) = p-3. \end{align*}$$

By Remark 4.6, the tensor product $L(\lambda ) \otimes L(\mu )$ has a composition factor of highest weight

$$\begin{align*}\nu = \lambda+\mu - \alpha_1 = (p-3) \cdot \varpi_1 + (b+b^\prime+1) \cdot \varpi_2 , \end{align*}$$

and by weight considerations, this implies that $T(\nu )$ is a direct summand of $L(\lambda ) \otimes L(\mu )$ . If $b+b^\prime \neq 0$ then we have $\nu \in C_1$ and $T(\nu )$ is nonsimple, contradicting the complete reducibility of $L(\lambda ) \otimes L(\mu )$ . We conclude that $b+b^\prime = 0$ , and $\lambda = a \varpi _1$ and $\mu = a^\prime \varpi _1$ with $a+a^\prime = p-1$ , as required.

Now we are ready to give the proof of our classification of pairs of simple G-modules whose tensor product is completely reducible, for $G = \mathrm {SL}_3(\Bbbk )$ .

4.5 Proofs: multiplicity freeness

In this subsection, we give the proof of Theorem 4.3; see page 27. We start with preliminary results about weight space dimensions and Verlinde coefficients.

Proof. As $\lambda ,\mu \in C_0$ and $\lambda + \mu \in \widehat {C}_0$ , we have $T(\lambda ) = \nabla (\lambda )$ and $T(\mu ) = \nabla (\mu )$ , and $T(\lambda ) \otimes T(\mu )$ is a direct sum of tilting modules $T(\nu ) = \nabla (\nu )$ with highest weights $\nu $ in the upper closure of $C_0$ . This implies that

$$\begin{align*}c_{\lambda,\mu}^\nu = [ T(\lambda) \otimes T(\mu) : T(\nu) ]_\oplus = [ \nabla(\lambda) \otimes \nabla(\mu) : \nabla(\nu) ]_\nabla \end{align*}$$

for all $\nu \in C_0 \cap X$ . A straightforward computation at the level of characters shows that

$$\begin{align*}[ \nabla(\varpi_1+\varpi_2) \otimes \nabla(\varpi_1+\varpi_2) : \nabla(\varpi_1+\varpi_2) ]_\nabla = 2 , \end{align*}$$

and since $a,a^\prime ,b,b^\prime \geq 1$ , Lemma 3.19 yields

$$\begin{align*}c_{\lambda,\mu}^{\lambda+\mu-\alpha_{\mathrm{h}}} = [ \nabla(\lambda) \otimes \nabla(\mu) : \nabla(\lambda+\mu-\alpha_{\mathrm{h}}) ]_\nabla \geq [ \nabla(\varpi_1+\varpi_2) \otimes \nabla(\varpi_1+\varpi_2) : \nabla(\varpi_1+\varpi_2) ]_\nabla = 2 , \end{align*}$$

as claimed.

Before giving the proof of Theorem 4.3, we record two corollaries of Lemma 4.14.

Proof. If $\lambda =0$ or $\mu =0$ then the claim is trivially satisfied. Now suppose that $\lambda \neq 0$ and $\mu \neq 0$ , and observe that $\lambda + \mu \in \widehat {C}_0$ implies that $\lambda \in C_0$ and $\mu \in C_0$ , in particular $L(\lambda ) = T(\lambda )$ and $L(\mu ) = T(\mu )$ . If $a,a^\prime ,b,b^\prime \geq 1$ then

$$\begin{align*}[ L(\lambda) \otimes L(\mu) : L(\lambda+\mu-\alpha_{\mathrm{h}}) ] \geq [ T(\lambda) \otimes T(\mu) : T(\lambda+\mu-\alpha_{\mathrm{h}}) ]_\oplus = c_{\lambda,\mu}^{\lambda+\mu-\alpha_{\mathrm{h}}} \geq 2 \end{align*}$$

by Lemma 4.14, and we conclude that $L(\lambda ) \otimes L(\mu )$ is multiplicity free only if $0 \in \{a,b,a^\prime ,b^\prime \}$ .

Proof. We assume that $a^\prime ,b^\prime \geq 1$ and show that $L(\lambda ) \otimes L(\mu )$ is multiplicity free only if $a+b=p-1$ . Since $\mu $ is reflection small with respect to $\lambda $ , we have $a+a'+b' \leq p-1$ and $b+a'+b' \leq p-1$ by Example 4.1 and $\lambda +\mu \in \widehat {C}_1$ . The assumption that $a^\prime ,b^\prime \geq 1$ further implies that $\lambda +\mu -\alpha _{\mathrm {h}} \in C_1$ and

$$ \begin{align*} a'+b' \leq 2 \cdot (a'+b') - 2 \leq 2 \cdot (a'&+b') - 2 + (a+b) - (p-1) \\ & = (a+a'+b') + (b+a'+b') - (p+1) \leq p-3 , \qquad \end{align*} $$

whence $\mu \in C_0$ . Consider the weights $\lambda _0 = s\boldsymbol {\cdot }\lambda \in C_0$ and $\nu = s \boldsymbol {\cdot } (\lambda + \mu - \alpha _{\mathrm {h}}) = \lambda _0 + w_0(\mu ) + \alpha _{\mathrm {h}} \in C_0$ (note that $w_0 = s_{\alpha _{\mathrm {h}}}$ ), and observe that

$$\begin{align*}\lambda_0 = (p-2-b) \cdot \varpi_1 + (p-2-a) \cdot \varpi_2 , \qquad \nu = (p-1-b-b^\prime) \cdot \varpi_1 + (p-1-a-a^\prime) \cdot \varpi_2 , \end{align*}$$

where $a+a^\prime < p-1$ and $b+b^\prime <p-1$ because $\mu $ is reflection small with respect to $\lambda $ . If $a+b>p-1$ then we have $\lambda _0+\alpha _{\mathrm {h}} \in \widehat {C}_0$ , and using Lemmas 3.10 and 4.14, we compute

$$\begin{align*}c_{\lambda_0,\mu}^\nu = c_{\nu,-w_0(\mu)}^{\lambda_0} \geq 2 \end{align*}$$

because $\lambda _0 = \nu - w_0(\mu ) - \alpha _{\mathrm {h}}$ . Now Lemma 3.11 implies that $L(\lambda ) \otimes L(\mu ) = L(s\boldsymbol {\cdot }\lambda _0) \otimes L(\mu )$ is not multiplicity free, and we conclude that $L(\lambda ) \otimes L(\mu )$ is multiplicity free only if $a+b=p-1$ .

Now we are ready to give the proof of our classification of pairs of simple G-modules whose tensor product is multiplicity free, for $G = \mathrm {SL}_3(\Bbbk )$ .