Semisimplification of the category of tilting modules for GL_n

Abstract

We describe the semisimplification of the monoidal category of tilting modules for the algebraic group $G{L}_n$ in characteristic $p>0$. In particular, we compute the dimensions of the indecomposable tilting modules modulo p.

Introduction

Let $\mathbb{k}$ be an algebraically closed field of characteristic $p\ge 0$ and $G_n$ denote the algebraic group $G{L}_n(\mathbb{k})$ for $n\ge 0$. The symmetric tensor category $\mathcal{R}ep(G_n)$ of finite-dimensional rational representations of $G_n$ is a lower finite highest weight category with irreducible, standard, costandard and indecomposable tilting modules $L_n(\lambda )$, ${\mathrm{\Delta }}_n(\lambda )$, ${\mathrm{\nabla }}_n(\lambda )$ and $T_n(\lambda )$ parametrized by their highest weight λ. In the usual coordinates, the dominant weight λ appearing here may be identified with an element of the poset

$$X_n^{+}=\{\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {\mathbb{Z}}^n|{\lambda }_1\ge \cdots \ge {\lambda }_n\}$$

ordered by the usual dominance ordering ⊴. Let $\mathcal{T}ilt(G_n)$ be the full subcategory of $\mathcal{R}ep(G_n)$ consisting of all tilting modules, which is a Karoubian rigid symmetric monoidal category. The defining n-dimensional representation $V_n$ of $G_n$ is an indecomposable tilting module, as are all of its (irreducible) exterior powers and their duals. These modules generate $\mathcal{T}ilt(G_n)$ as a Karoubian monoidal category (i.e., taking tensor products, direct sums and direct summands).

The semisimplification

$$\overline{\mathcal{T}ilt(G_n)}:=\mathcal{T}ilt(G_n)/\mathcal{N}$$

of the category $\mathcal{T}ilt(G_n)$ is its quotient by the tensor ideal $\mathcal{N}$ consisting of all negligible morphisms. This is a semisimple symmetric tensor category with irreducible objects arising from the indecomposable tilting modules whose dimension is non-zero modulo p; see [15] for further discussion and historical remarks. Of course, if $p=0$ the category $\mathcal{R}ep(G_n)$ is already semisimple so coincides with the semisimplification $\overline{\mathcal{T}ilt(G_n)}$, and the irreducible objects in $\overline{\mathcal{T}ilt(G_n)}$ are labeled by the set $X_{n,0}^{+}:=X_n^{+}$ of all dominant weights. The case $p\ge n$ may also be regarded as classical: in this case, the category $\overline{\mathcal{T}ilt(G_n)}$ is the so-called Verlinde category, with irreducible objects arising from the indecomposable tilting modules of highest weight belonging to the set

$$X_{n,p}^{+}:=\{\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in X_n^{+}|{\lambda }_1-{\lambda }_n<p-n+1\},$$

interpreting $X_{0,p}^{+}$ as $\{\varnothing \}$. The classical proof of this from [17], [18] goes as follows. As $X_{n,p}^{+}$ is the fundamental alcove, the linkage principle implies that $T_n(\lambda )={\mathrm{\Delta }}_n(\lambda )$ for λ in the upper closure ${\bar{X}}_{n,p}^{+}$ (defined by replacing < in (1.3) by ≤). By the Weyl dimension formula, it follows that $T_n(\lambda )$ is of non-zero dimension modulo p for $\lambda \in X_{n,p}^{+}$, and its identity morphism is negligible for $\lambda \in {\bar{X}}_{n,p}^{+}\setminus X_{n,p}^{+}$. Then an argument with translation functors gives that the identity morphism of $T_n(\lambda )$ is negligible for any $\lambda \in X_n^{+}\setminus X_{n,p}^{+}$, hence, these modules are all of dimension zero modulo p.

In this article, we treat the remaining situations when $0<p<n$. Note that the case $p=2$ was worked out already in [15, §8]. To formulate the main result in general, assume that $n,p>0$ and let

$$n=n_0+n_{1}p+\cdots +n_r{p}^r$$

be the p-adic decomposition of n, so $0\le n_0,\dots ,n_{r-1}<p$ and $0<n_r<p$. We define an embedding

$$ı:X_{n_0}^{+}\times X_{n_1}^{+}\times \cdots \times X_{n_r}^{+}\hookrightarrow X_n^{+}$$

sending $\underset{\_}{\lambda }=({\lambda }^{(0)},\dots ,{\lambda }^{(r)})$ to the dominant conjugate of the n-tuple that is the concatenation ${\lambda }^{(0)}\bigsqcup \underset{\mathit{\text{p}}\text{copies}}{\underbrace{{\lambda }^{(1)}\bigsqcup \cdots \bigsqcup {\lambda }^{(1)}}}\bigsqcup \underset{p^2\text{ copies}}{\underbrace{{\lambda }^{(2)}\bigsqcup \cdots \bigsqcup {\lambda }^{(2)}}}\bigsqcup \cdots \bigsqcup \underset{p^r\text{ copies}}{\underbrace{{\lambda }^{(r)}\bigsqcup \cdots \bigsqcup {\lambda }^{(r)}}}$. Let

$$X_{n,p}^{+}:=ı(X_{n_0,p}^{+}\times \cdots \times X_{n_r,p}^{+})\subset X_n^{+}.$$

See (5.3)–(5.4) below for a more conceptual description of this set. Also let ⊠ be the Deligne tensor product of tensor categories (e.g., see [14, §4.6]). The Deligne tensor product of semisimple symmetric tensor categories is again a semisimple symmetric tensor category.

Main Theorem

For $p>0$ as above, there is a symmetric monoidal equivalence

$${\mathit{\Xi }}_n:\overline{\mathcal{T}ilt(G_{n_0})}⊠\cdots ⊠\overline{\mathcal{T}ilt(G_{n_r})}\to \overline{\mathcal{T}ilt(G_n)}$$

sending $T_{n_0}({\lambda }^{(0)})⊠\cdots ⊠T_{n_r}({\lambda }^{(r)})$ for $\underset{\_}{\lambda }=({\lambda }^{(0)},\dots ,{\lambda }^{(r)})\in X_{n_0,p}^{+}\times \cdots \times X_{n_r,p}^{+}$ to $T_n(ı(\underset{\_}{\lambda }))$. In particular, the irreducible objects of $\overline{\mathcal{T}ilt(G_n)}$ are the indecomposable tilting modules with highest weight in $X_{n,p}^{+}$.

Example

If $p=5$ and $n=13=3+2\cdot 5$, this implies that $\overline{\mathcal{T}ilt(G_{13})}$ is equivalent to $\overline{\mathcal{T}ilt(G_3)}⊠\overline{\mathcal{T}ilt(G_2)}$. The bijection $ı:X_{3,5}^{+}\times X_{2,5}^{+}\to X_{13,5}^{+}$ between the labeling sets takes $\underset{\_}{\lambda }=({\lambda }^{(0)},{\lambda }^{(1)})\in X_3^{+}\times X_2^{+}$ with ${\lambda }_1^{(0)}-{\lambda }_3^{(0)}<3$ and ${\lambda }_1^{(1)}-{\lambda }_2^{(1)}<4$ to

$$ı(\underset{\_}{\lambda })={({\lambda }_1^{(0)},{\lambda }_2^{(0)},{\lambda }_3^{(0)},{\lambda }_1^{(1)},{\lambda }_2^{(1)},{\lambda }_1^{(1)},{\lambda }_2^{(1)},{\lambda }_1^{(1)},{\lambda }_2^{(1)},{\lambda }_1^{(1)},{\lambda }_2^{(1)},{\lambda }_1^{(1)},{\lambda }_2^{(1)})}^{+}\in X_{13}^{+}$$

where + denotes dominant conjugate. So ${\mathrm{\Xi }}_{13}(V_3⊠\mathbb{k})\cong V_{13}$, ${\mathrm{\Xi }}_{13}(\mathbb{k}⊠V_2)\cong {\bigwedge }^5{V}_{13}$ and ${\mathrm{\Xi }}_{13}(V_3⊠V_2)\cong {\bigwedge }^6{V}_{13}\cong V_{13}\otimes {\bigwedge }^5{V}_{13}$ (isomorphisms in $\overline{\mathcal{T}ilt(G_{13})}$).

Corollary

If $\lambda \in X_n^{+}\setminus X_{n,p}^{+}$ then $\dim T_n(\lambda )\equiv 0(\mathrm{mod}p)$. If $\lambda \in X_{n,p}^{+}$, so that $\lambda =ı(\underset{\_}{\lambda })$ for $\underset{\_}{\lambda }=({\lambda }^{(0)},\dots ,{\lambda }^{(r)})\in X_{n_0,p}^{+}\times \cdots \times X_{n_r,p}^{+}$, then we have that

$$\dim T_n(\lambda )\equiv \prod _{i=0}^r\dim {\mathit{\Delta }}_{n_i}({\lambda }^{(i)})(\mathrm{mod}p).$$

The right hand side here may be computed explicitly using the Weyl dimension formula.

Proof

For each $i=0,\dots ,r$, we have that $p>n_i$, so by the classical description of Verlinde categories we have that $\dim T_{n_i}({\lambda }^{(i)})\equiv \dim {\mathrm{\Delta }}_{n_i}({\lambda }^{(i)})(\mathrm{mod}p)$ for ${\lambda }^{(i)}\in X_{n_i}^{+}$. Now the corollary follows from the theorem since symmetric monoidal functors are trace-preserving, hence, they also respect categorical dimensions. □

The Main Theorem gives rise to a categorification of Lucas' theorem in the following sense. If $k=k_0+k_{1}p+\cdots +k_r{p}^r$ for $0\le k_0,\dots ,k_r<p$, then ${\bigwedge }^k{V}_n\in \overline{\mathcal{T}ilt(G_n)}$ is the image of the irreducible object ${\bigwedge }^{k_0}{V}_{n_0}⊠\cdots ⊠{\bigwedge }^{k_r}{V}_{n_r}\in \overline{\mathcal{T}ilt(G_{n_0})}⊠\cdots ⊠\overline{\mathcal{T}ilt(G_{n_r})}$ under the equivalence ${\mathrm{\Xi }}_n$ from the theorem. We deduce on taking categorical dimensions that

$$\left(\begin{array}{c}n\\ k\end{array}\right)\equiv \prod _{i=0}^r\left(\begin{array}{c}{n}_i\\ k_i\end{array}\right)(\mathrm{mod}p),$$

which is exactly the classical Lucas theorem.

An essential step in the proof is provided by a theorem of Donkin from [10], which gives a version of skew Howe duality for the general linear group. In fact, we rephrase Donkin's result in terms of what we call the Schur category; see Theorem 4.14 for the statement. The Schur category is a strict monoidal category closely related to the classical Schur algebra; see Definition 4.2. It also has an explicit diagrammatic realization in terms of webs, which is due to Cautis, Kamnitzer and Morrison [7]. Since we are working in positive characteristic, we have included a self-contained treatment establishing the connection between the Schur category and webs via an approach which is independent of [7]; see Theorem 4.10.

The Main Theorem reduces the study of $\overline{\mathcal{T}ilt(G_n)}$ for all $p\ge 0$ to the classical cases in which $p=0$ or $p>n$. In these classical cases, it can be helpful to think about the combinatorial structure of $\overline{\mathcal{T}ilt(G_n)}$ from the perspective of categorification. Let $\mathfrak{s}$ be the affine Kac-Moody algebra ${\mathfrak{sl}}_{\infty }$ if $p=0$ or ${\widehat{\mathfrak{sl}}}_p$ if $p>n$, with fundamental weights ${\mathrm{\Lambda }}_i$ and simple coroots $h_i$ for $i\in \mathbb{Z}/p\mathbb{Z}$. There is a well-known categorical action making $\mathcal{R}ep(G_n)$ into a 2-representation of the Kac-Moody 2-category $\mathfrak{U}(\mathfrak{s})$. (The quickest way to construct this is to apply [6, Theorem 4.11], starting from the action of the degenerate Heisenberg category of central charge zero under which ↑ acts by tensoring with $V_n$ and ↓ acts by tensoring with $V_n^{⁎}$, as is discussed in the introduction of [6].) This categorical action restricts to give an action of $\mathfrak{U}(\mathfrak{s})$ on $\mathcal{T}ilt(G_n)$ such that

$$\mathbb{C}{\otimes }_{\mathbb{Z}}{K}_0(\mathcal{T}ilt(G_n))\cong {\bigwedge }^n{\mathrm{Nat}}_p$$

as an $\mathfrak{s}$-module, where ${\mathrm{Nat}}_p$ is a natural level zero representation of $\mathfrak{s}$ with basis ${(m_i)}_{i\in \mathbb{Z}}$ such that $m_i$ is of weight ${\mathrm{\Lambda }}_{i-1}-{\mathrm{\Lambda }}_i$; see the discussion in the introduction of [3], or [23, Proposition 6.5]. In particular, $\mathbb{C}{\otimes }_{\mathbb{Z}}{K}_0(\mathcal{T}ilt(G_n))$ is generated as an $\mathfrak{s}$-module by the class $[\mathbb{k}]$ of the trivial module, which corresponds under (1.8) to the vector $m_0\wedge m_{-1}\wedge \cdots \wedge m_{1-n}\in {\bigwedge }^n{\mathrm{Nat}}_p$ of weight ${\mathrm{\Lambda }}_{-n}-{\mathrm{\Lambda }}_0$. The ideal $\mathcal{N}$ of negligible morphisms defines a sub-2-representation, hence, the quotient $\overline{\mathcal{T}ilt(G_n)}$ is a 2-representation as well. Its complexified Grothendieck ring satisfies

$$\mathbb{C}{\otimes }_{\mathbb{Z}}{K}_0(\overline{\mathcal{T}ilt(G_n)})\cong V({\mathrm{\Lambda }}_{-n}-{\mathrm{\Lambda }}_0),$$

i.e., it is the level zero extremal weight module parametrized by the minuscule weight ${\mathrm{\Lambda }}_{-n}-{\mathrm{\Lambda }}_0$ in the sense of [21]. This follows because, as an $\mathfrak{s}$-module, $\mathbb{C}{\otimes }_{\mathbb{Z}}{K}_0(\overline{\mathcal{T}ilt(G_n)})$ is generated by a vector of weight ${\mathrm{\Lambda }}_{-n}-{\mathrm{\Lambda }}_0$, and it is minuscule as all of its weights λ satisfy $〈h_i,\lambda 〉\in \{0,1,-1\}$ for all $i\in \mathbb{Z}/p\mathbb{Z}$. The latter assertion follows from the semisimplicity of the category $\overline{\mathcal{T}ilt(G_n)}$ by invoking some of the general structure theory of Kac-Moody 2-representations. In more detail, semisimplicity implies that the representation-theoretic Kashiwara operators ${\epsilon }_i,{\varphi }_i$ as defined e.g. in [6, §5.1] satisfy ${\epsilon }_i(L),{\varphi }_i(L)\le 1$ for all irreducible objects $L\in \overline{\mathcal{T}ilt(G_n)}$ and all $i\in \mathbb{Z}/p\mathbb{Z}$. Since the weight λ of the class of L in $\mathbb{C}{\otimes }_{\mathbb{Z}}{K}_0(\overline{\mathcal{T}ilt(G_n)})$ satisfies $〈h_i,\lambda 〉={\varphi }_i(L)-{\epsilon }_i(L)$ by [6, Lemma 5.2], this implies that $〈h_i,\lambda 〉\in \{0,1,-1\}$ for all i.

We remark finally that there is also a generalization of our Main Theorem to the quantum general linear group $G_{n,q}$ for any $q\in {\mathbb{k}}^{\times }$ such that $q^2$ is a primitive ℓth root of unity. It is related to the quantum Lucas theorem. The proof in the quantum case is quite similar, using Donkin's skew Howe duality established in [11] formulated in terms of the q-Schur category, which again can be viewed diagrammatically in terms of the webs of [7]. This will be developed in a subsequent paper.

Conventions

All categories will be $\mathbb{k}$-linear with finite-dimensional Hom-spaces, and all functors will be $\mathbb{k}$-linear. A category is Karoubian if it is additive and idempotent complete. Functors between Karoubian categories are automatically additive due to the assumption that they are $\mathbb{k}$-linear,