We introduce the alternating Schur algebra $AS_F(n,d)$ as the commutant of the action of the alternating group $A_d$ on the $d$-fold tensor power of an $n$-dimensional $F$-vector space. When $F$ has characteristic different from $2$, we give a basis of $AS_F(n,d)$ in terms of bipartite graphs, and a graphical interpretation of the structure constants. We introduce the abstract Koszul duality functor on modules for the even part of any $\mathbf Z/2\mathbf Z$-graded algebra. The algebra $AS_F(n,d)$ is $\mathbf Z/2\mathbf Z$-graded, having the classical Schur algebra $S_F(n,d)$ as its even part. This leads to an approach to Koszul duality for $S_F(n,d)$-modules that is amenable to combinatorial methods. We characterize the category of $AS_F(n,d)$-modules in terms of $S_F(n,d)$-modules and their Koszul duals. We use the graphical basis of $AS_F(n,d)$ to study the dependence of the behavior of derived Koszul duality on $n$ and $d$.
We give a proof of the parabolic/singular Koszul duality for the category O of affine Kac-Moody algebras. The main new tool is a relation between moment graphs and finite codimensional affine Schubert varieties. We apply this duality to q-Schur algebras and to cyclotomic rational double affine Hecke algebras.
We investigate the endotrivial modules for the Schur covers of the symmetric and alternating groups and determine the structure of their group of endotrivial modules in all characteristics. We provide a full description of this group by generators and relations in all cases.
Using combinatorics and character theory, we determine the imprimitive faithful complex characters, i.e., the irreducible faithful complex characters which are induced from proper subgroups, of the Schur covers of the symmetric and alternating groups. Furthermore, for every imprimitive character we establish all its minimal block stabilizers. As a corollary, we also determine the monomial faithful characters of the Schur covers.
We establish Schur-Weyl reciprocity for the q-analogue of the alternating group. We analyze the sign q-permutation representation of the Hecke algebra on the tensor product of the super vector space V in detail, and examine its restriction to the q-analogue of the alternating group. In consequence, we find out that if the dimensions of even part and odd part of V are same, then the centralizer of the q-analogue of the alternating group is a Z_2-crossed product of the centralizer of the Hecke algebra and obtain Schur-Weyl reciprocity between them. Though the structure of the centralizer is more…
Schur modules give the irreducible polynomial representations of the general linear group $\mathrm{GL}_t$. Viewing the symmetric group $\mathfrak{S}_t$ as a subgroup of $\mathrm{GL}_t$, we may restrict Schur modules to $\mathfrak{S}_t$ and decompose the result into a direct sum of Specht modules, the irreducible representations of $\mathfrak{S}_t$. We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for $\mathfrak{S}_t$ for $t$ large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternat…
This is a report on recent work of Chalupnik and Touze. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors.
Given a quasi-hereditary algebra $B$, we present conditions which guarantee that the algebra $\gr B$ obtained by grading $B$ by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that $B$ might possess. The method involves working with a pair $(A,{\mathfrak a})$ consisting of a quasi-hereditary algebra $A$ and a (positively) graded subalgebra $\mathfrak a$. The algebra $B$ arises as a quotient $B=A/J$ of $A$ by a defining ideal $J$ of $A$. Along the way, we also show that the standard (Weyl) modules for $B$ have …
We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related factorizations involving sums of two Schur polynomials, and certain odd-sized sets of variables. Our results generalize the factorization identities proved by Ciucu and Krattenthaler (Advances in combinatorial mathematics, 39-59, 2009) for partitions of rectangular shape. We observe that if, in some of the results, the partitions are taken to have rectangul…
For any finite group G it is an interesting question to ask which ordinary irreducible representations of G remain irreducible in a given characteristic p. We answer this question for p=2 when G is the proper double cover of the alternating group. As a key ingredient in the proof, we prove a formula for the decomposition numbers in Rouquier blocks of double covers of symmetric groups, in terms of Schur P-functions.
We define a graded quasi-hereditary covering for the cyclotomic quiver Hecke algebras $\mathcal{R}^Λ_n$ of type $A$ when $e=0$ (the linear quiver) or $e\ge n$. We show that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When $e=0$ we show that the KLR grading on the quiver Hecke algebras is compatible with the gradings on parabolic category $\mathcal{O}$ previously introduced in the works of Beilinson, Ginzburg and Soergel and Backelin. As a consequence, we show that when $e=0$ our graded Schur algebras are Koszul over field of chara…
In this paper we consider the cohomology of four groups related to the virtual braids of [Kauffman] and [Goussarov-Polyak-Viro], namely the pure and non-pure virtual braid groups (PvB_n and vB_n, respectively), and the pure and non-pure flat braid groups (PfB_n and fB_n, respectively). The cohomologies of PvB_n and PfB_n admit an action of the symmetric group S_n. We give a description of the cohomology modules H^i(PvB_n,Q) and H^i(PfB_n,Q) as sums of S_n-modules induced from certain one-dimensional representations of specific subgroups of S_n. This in particular allows us to conclude that H^i…
Let $U_ζ$ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra $\mathfrak g$ and a root of unity $ζ$. When $L,L'$ are irreducible $U_ζ$-modules having regular highest weights, the dimension of $\operatorname{Ext}^n_{U_ζ}(L,L')$ can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of $U_ζ$. This paper shows for $L,L'$ irreducible modules in a singular block that $\dim\operatorname{Ext}^n_{U_ζ}(L,L')$ is explicitly determined using the coefficients of parabolic Kazhdan-Lusztig polynomials. …
The partition algebra $\mathsf{P}_k(n)$ and the symmetric group $\mathsf{S}_n$ are in Schur-Weyl duality on the $k$-fold tensor power $\mathsf{M}_n^{\otimes k}$ of the permutation module $\mathsf{M}_n$ of $\mathsf{S}_n$, so there is a surjection $\mathsf{P}_k(n) \to \mathsf{Z}_k(n) := \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k}),$ which is an isomorphism when $n \ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\mathsf{Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities …
There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation theory of the symmetric group and representation theory of the algebraic supergroup $Q(n)$ via appropriate Schur (super)algebras and Schur functors. The second approach follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras and connects projective representation theory of symmetric groups in characteristic $p$ to the crystal graph of th…
