Schur Functor Research Articles

On Noetherian algebras, Schur functors and Hemmer–Nakano dimensions

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander’s correspondence, classical Schur–Weyl duality and Soergel’s Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B B , and module categories of endomorphism algebras of generator-relative cogenerators over B B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n n so that the endomorphism algebra of a generator-cogenerator provides an n n -faithful cover, in the sense of Rouquier, of B B . The degree n n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B B is bounded above by the number of non-isomorphic simple modules of B B . We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category O \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category O \mathcal {O} establishing an integral version of Soergel’s Struktursatz. We show that deformations of the combinatorial Soergel’s functor have better homological properties than the classical one.

Modular plethystic isomorphisms for two-dimensional linear groups

Let E be the natural representation of the special linear group SL2(K) over an arbitrary field K. We use the two dual constructions of the symmetric power when K has prime characteristic to construct an explicit isomorphism SymmSymℓE≅SymℓSymmE. This generalises Hermite reciprocity to arbitrary fields. We prove a similar explicit generalisation of the classical Wronskian isomorphism, namely SymmSymℓE≅⋀mSymℓ+m−1E. We also generalise a result first proved by King, by showing that if ∇λ is the Schur functor for the partition λ and λ∘ is the complement of λ in a rectangle with ℓ+1 rows, then ∇λSymℓE≅∇λ∘SymℓE. To illustrate that the existence of such ‘plethystic isomorphisms’ is far from obvious, we end by proving that the generalisation ∇λSymℓE≅∇λ′Symℓ+ℓ(λ′)−ℓ(λ)E of the Wronskian isomorphism, known to hold for a large class of partitions over the complex field, does not generalise to fields of prime characteristic, even after considering all possible dualities.

A COMBINATORIAL TRANSLATION PRINCIPLE AND DIAGRAM COMBINATORICS FOR THE GENERAL LINEAR GROUP

Let k be an algebraically closed field of characteristic p > 0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the general linear group over k in terms of cap diagrams under the assumption that p is bigger than the greatest hook length in the partitions involved. Then we introduce and study the rational Schur functor from a category of GLn-modules to the category of modules for the walled Brauer algebra. As a corollary, we obtain the decomposition numbers for the walled Brauer algebra when p is bigger than the greatest hook length in the partitions involved. This is a sequel to an earlier paper on the symplectic group and the Brauer algebra.

When the Schur functor induces a triangle-equivalence between Gorenstein defect categories

Let $R$ be an Artin algebra and $e$ an idempotent of $R$. Assume that ${\rm Tor}_i^{eRe}(Re,G)=0$ for any $G\in{\rm GProj} eRe$ and $i$ sufficiently large. Necessary and sufficient conditions are given for the Schur functor $S_e$ to induce a triangle-equivalence $\mathbb{D}_{def}(R)\simeq\mathbb{D}_{def}(eRe)$. Combine this with a result of Psaroudakis-Skartsaterhagen-Solberg [29], we provide necessary and sufficient conditions for the singular equivalence $\mathbb{D}_{sg}(R)\simeq\mathbb{D}_{sg}(eRe)$ to restrict to a triangle-equivalence $\underline{{\rm GProj} R}\simeq\underline{{\rm GProj} eRe}$. Applying these to the triangular matrix algebra $T=\left( \begin{array}{cc} A & M \quad 0 & B \end{array} \right)$, corresponding results between candidate categories of $T$ and $A$ (resp. $B$) are obtained. As a consequence, we infer Gorensteinness and CM-freeness of $T$ from those of $A$ (resp. $B$). Some concrete examples are given to indicate one can realise the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algabras.

The image of the Specht module under the inverse Schur functor in arbitrary characteristic

This paper gives a necessary and sufficient condition for the image of the Specht module under the inverse Schur functor to be isomorphic to the dual Weyl module in characteristic 2, and gives an elementary proof that this isomorphism holds in all cases in all other characteristics. These results are new in characteristics 2 and 3. We deduce some new examples of indecomposable Specht modules in characteristic 2. When the isomorphism does not hold, the dual Weyl module is still a quotient of the image of the Specht module, and we prove some additional results: we demonstrate that the image need not have a filtration by dual Weyl modules, we bound the dimension of the kernel of the quotient map, and we give some explicit descriptions for particular cases. Our method is to view the Specht and dual Weyl modules as quotients of suitable exterior powers by the Garnir relations.

Deligne Categories and the Periplectic Lie Superalgebra

We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\mathfrak{p}(n)$ as $n \to \infty$. The paper gives a construction of the tensor category $Rep(\underline{P})$, possessing nice universal properties among tensor categories over the category $\mathtt{sVect}$ of finite-dimensional complex vector superspaces. First, it is the envelope of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. Secondly, given a tensor category $\mathcal{C}$ over $\mathtt{sVect}$, exact tensor functors $Rep(\underline{P})\longrightarrow \mathcal{C}$ classify pairs $(X, \omega)$ in $\mathcal{C}$ where $\omega: X \otimes X \to \Pi \mathbf{1}$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor. The category $Rep(\underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $Rep(\mathfrak{p}(n))$ ($n\geq 1$) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes $Rep(\underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $\mathtt{sVect} \boxtimes Rep(\underline{GL}_t)$. An upcoming paper by the authors will give results on the abelian and tensor structure of $Rep(\underline{P})$.

A Characterisation of Morita Algebras in Terms of Covers

A pair (A, P) is called a cover of EndA(P)op if the Schur functor HomA(P,−) is fully faithful on the full subcategory of projective A-modules, for a given projective A-module P. By definition, Morita algebras are the covers of self-injective algebras and then P is a faithful projective-injective module. Conversely, we show that A is a Morita algebra and EndA(P)op is self-injective whenever (A, P) is a cover of EndA(P)op for a faithful projective-injective module P.

Plethysms of symmetric functions and representations of SL 2 (C)

Let ∇λ denote the Schur functor labelled by the partition λ and let E be the natural representation of SL2(C). We make a systematic study of when there is an isomorphism ∇λSymlE≅∇μSymmE of representations of SL2(C). Generalizing earlier results of King and Manivel, we classify all such isomorphisms when λ and μ are conjugate partitions and when one of λ or μ is a rectangle. We give a complete classification when λ and μ each have at most two rows or columns or is a hook partition and a partial classification when l=m. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when ∇λSymlE is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon’s enumeration of plane partitions, and prove a new q-binomial identity in this setting

Deligne Categories and the Limit of Categories Rep(GL(m|n))

Abstract For each integer $t$ a tensor category $\mathcal{V}_t$ is constructed, such that exact tensor functors $\mathcal{V}_t\rightarrow \mathcal{C}$ classify dualizable $t$-dimensional objects in $\mathcal{C}$ not annihilated by any Schur functor. This means that $\mathcal{V}_t$ is the “abelian envelope” of the Deligne category $\mathcal{D}_t=\operatorname{Rep}(GL_t)$. Any tensor functor $\operatorname{Rep}(GL_t)\longrightarrow \mathcal{C}$ is proved to factor either through $\mathcal{V}_t$ or through one of the classical categories $\operatorname{Rep}(GL(m|n))$ with $m-n=t$. The universal property of $\mathcal{V}_t$ implies that it is equivalent to the categories $\operatorname{Rep}_{\mathcal{D}_{t_1}\otimes \mathcal{D}_{t_2}}(GL(X),\epsilon )$, ($t=t_1+t_2$, $t_1$ not an integer) suggested by Deligne as candidates for the role of abelian envelope.

RELATING TENSOR STRUCTURES ON REPRESENTATIONS OF GENERAL LINEAR AND SYMMETRIC GROUPS

For polynomial representations of GL n of a fixed degree, H. Krause defined a new “internal tensor product” using the language of strict polynomial functors. We show that over an arbitrary commutative base ring k, the Schur functor carries this internal tensor product to the usual Kronecker tensor product of symmetric group representations. This is true even at the level of derived categories. The new tensor product is a substantial enrichment of the Kronecker tensor product. E.g., in modular representation theory it brings in homological phenomena not visible on the symmetric group side. We calculate the internal tensor product over any commutative ring k in several interesting cases involving classical functors and the Weyl functors. We show an application to the Kronecker problem in characteristic zero when one partition has two rows or is a hook.

Connectedness of cup products for polynomial representations of GLn and applications

We find conditions such that cup products induce isomorphisms in low degrees for extensions between stable polynomial representations of the general linear group. We apply this result to prove generalizations and variants of the Steinberg tensor product theorem. Our connectedness bounds for cup product maps depend on numerical invariants which seem also relevant to other problems, for example to study the cohomological behavior of the Schur functor.

Cohomology of automorphism groups of free groups with twisted coefficients

We compute the groups H^*(mathrm {Aut}(F_n); M) and H^*(mathrm {Out}(F_n); M) in a stable range, where M is obtained by applying a Schur functor to H_mathbb {Q} or H^*_mathbb {Q}, respectively the first rational homology and cohomology of F_n. The answer may be described in terms of stable multiplicities of irreducibles in the plethysm mathrm {Sym}^k circ mathrm {Sym}^l of symmetric powers. We also compute the stable integral cohomology groups of mathrm {Aut}(F_n) with coefficients in H or H^*.

Some remarks on Gill's theorems on Young modules

In a recent paper, [10], Gill, using methods of the modular representation theory of finite groups, describes some results on the tensor product of Young modules for symmetric groups. We here give an alternative approach using the polynomial representation theory of general linear groups and the Schur functor. The main result is a formula for the multiplicities of Young modules in a tensor product in terms of the characters of the simple polynomial modules for general linear groups. Our approach is also valid for Young modules for Hecke algebras of type A at roots of unity and here the formula involves the characters of the simple polynomial modules for quantised general linear groups.

A geometric Schur functor

We give geometric descriptions of the category \(C_k(n,d)\) of rational polynomial representations of \(GL_n\) over a field \(k\) of degree \(d\) for \(d\le n\), the Schur functor and Schur–Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category \(C_k(n,d)\) and the Schur functor.

ON THE SLOPE OF THE SCHUR FUNCTOR OF A VECTOR BUNDLE

We prove that, for any complex vector bundle E of rank e on a compact Kahler manifold X, we have that μ(SE) = |λ| μ(E) for any λ = (λ1, ..., λe−1) with λi ∈ N and λ1 ≥ ... ≥ λe−1, where |λ| = λ1 + ... + λe−1, the symbol S denotes the Schur functor and μ is the slope. This result has already been stated, without proof, by Ottaviani in 1995. AMS Subject Classification: 19L10, 55R10

On admissible tensor products inp-adic Hodge theory

AbstractWe prove that ifWandW′are non-zeroB-pairs whose tensor product is crystalline (or semi-stable or de Rham or Hodge–Tate), then there exists a characterμsuch thatW(μ−1) andW′(μ) are crystalline (or semi-stable or de Rham or Hodge–Tate). We also prove that ifWis aB-pair and ifFis a Schur functor (for example Symnor Λn) such thatF(W) is crystalline (or semi-stable or de Rham or Hodge–Tate) and if the rank ofWis sufficiently large, then there is a characterμsuch thatW(μ−1) is crystalline (or semi-stable or de Rham or Hodge–Tate). In particular, these results apply top-adic representations.

The Schur functor on tensor powers

Let M be a left module for the Schur algebra S(n, r), and let \({s \in \mathbb{Z}^+}\) . Then \({M^{\otimes s}}\) is a \({(S(n,\,rs), F{\mathfrak{S}_{s}})}\) -bimodule, where the symmetric group \({{\mathfrak{S}_s}}\) on s letters acts on the right by place permutations. We show that the Schur functor f rs sends \({M^{\otimes s}}\) to the \({(F{\mathfrak{S}_{rs}},F{\mathfrak{S}_s})}\) -bimodule \({F\mathfrak{S}_{rs}\otimes_{F(\mathfrak{S}_{r}\wr{\mathfrak{S}_s})} ((f_rM)^{\otimes s}\otimes_{F} F{\mathfrak{S}_s})}\) . As a corollary, we obtain the image under the Schur functor of the Lie power L s(M), exterior power \({\bigwedge^s(M)}\) of M and symmetric power S s(M).

Semisimplicity in symmetric rigid tensor categories

Let λ be a partition of a positive integer n. Let C be a symmetric rigid tensor category over a field k of characteristic 0 or char ( k ) > n , and let V be an object of C . In our main result (Theorem 4.3) we introduce a finite set of integers F ( λ ) and prove that if the Schur functor S λ V of V is semisimple and the dimension of V is not in F ( λ ) , then V is semisimple. Moreover, we prove that for each d ∈ F ( λ ) there exist a symmetric rigid tensor category C over k and a non-semisimple object V ∈ C of dimension d such that S λ V is semisimple (which shows that our result is the best possible). In particular, Theorem 4.3 extends two theorems of Serre for C = Rep ( G ) , G is a group, and S λ V is ⋀ n V or Sym n V , and proves a conjecture of Serre (1997) [S2].

On decomposition numbers with Jantzen filtration of cyclotomic 𝑞-Schur algebras

Let S ( Λ ) \mathscr {S}(\varLambda ) be the cyclotomic q q -Schur algebra associated to the Ariki-Koike algebra H n , r \mathscr {H}_{n,r} , introduced by Dipper, James and Mathas. In this paper, we consider v v -decomposition numbers of S ( Λ ) \mathscr {S}(\varLambda ) , namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a v v -analogue of the result obtained by Shoji and Wada, a product formula for v v -decomposition numbers of S ( Λ ) \mathscr {S}(\varLambda ) , which asserts that certain v v -decomposition numbers are expressed as a product of v v -decomposition numbers for various cyclotomic q q -Schur algebras associated to Ariki-koike algebras H n i , r i \mathscr {H}_{n_i,r_i} of smaller rank. Moreover, we prove a similar formula for v v -decomposition numbers of H n , r \mathscr {H}_{n,r} by using a Schur functor.

Additive Families of GLn-Invariants of Schur Functors

We extend the symbolic method of [5] to invariants for the action of the general linear group on any Schur or co-Schur functor. We apply this symbolic method to prove that certain families of invariants of the Schur (co-Schur) functors are additive.