arXiv subjects
Publications and source records attributed to Shraddha Srivastava.
We provide a categorification for the hybrid Kazhdan-Lusztig bases of the Hecke algebra by constructing and investigating a series of new stratified structures on the principal block of BGG category $\mathcal{O}$.
Let $\mathcal{H} = \mathcal{H}(W,S)$ be the Hecke algebra of the Coxeter system $(W,S)$ over $\mathbb{Z}[q^{\pm1}]$, where $W$ is the Weyl group of a symmetrizable Kac-Moody algebra. In this paper, we show that the matrix of Kazhdan-Lusztig polynomials of $\mathcal{H}$ factorizes into a product of $|S|$ many matrices, each of which has entries as polynomials in $q$ with nonnegative coefficients. To achieve this goal, we use hybrid basis $TC^J$ for $J\subseteq S$ of $\mathcal{H}$, defined by Grojnowski-Haiman. The intermediate matrices in the aforementioned factorization turn out to be the transition matrices from $TC^J$-basis to $TC^I$-basis for $I\subset J$. Equivalently, these coefficients can be computed using a natural restriction map from $\mathcal{H}$ to the parabolic Hecke algebra $\mathcal{H}_J$. Moreover, following the ideas from Grojnowski-Haiman, we also give a geometric proof of the positivity of these coefficients.
Let $\mathcal{P}_k(δ)$, where $k$ is a positive integer and $δ$ some complex parameter, be the classical partition algebra over the complex numbers. In the case when $δ=n$, it is well-known that the algebra $\mathcal{P}_k(δ)$ is the centralizer of the symmetric group $S_n$ acting on the $k$-fold tensor space of the natural representation of $S_n$, for $n\geq 2k$. The algebra $\mathcal{P}_k(δ)$ is semi-simple for generic values of $δ$. In this paper, we show that semi-simple partition algebras appear as the centralizer algebras for certain representations of the rook monoids given by an iterative restriction-induction of the trivial representation. Along the way, we also give a decomposition of this iterative representation of the rook monoid into various tensor spaces and show that the corresponding dimensions are given by generalized Bell numbers.
We study analogues of Kronecker coefficients for symmetric inverse semigroups, for dual symmetric inverse semigroups and for the inverse semigroups of bijections between subquotients of finite sets. In all cases we reduce the problem of determination of such coefficients to some group-theoretic and combinatorial problems. For symmetric inverse semigroups, we provide an explicit formula in terms of the classical Kronecker and Littlewood--Richardson coefficients for symmetric groups.
We give a complete combinatorial classification of those parabolic Verma modules in the principal block of the parabolic category $\mathcal{O}$ associated to a minimal or a maximal parabolic subalgebra of the special linear Lie algebra for which the answer to Kostant's problem is positive.
We use character polynomials to obtain a positive combinatorial interpretation of the multiplicity of the sign representation in irreducible polynomial representations of $GL_n(\mathbb{C})$ indexed by two-column and hook partitions. Our method also yields a positive combinatorial interpretation for the multiplicity of the trivial representation of $S_n$ in an irreducible polynomial representation indexed by a hook partition.
We introduce the multiset partition algebra $\mathcal{MP}_k(ξ)$ over $F[ξ]$, where $F$ is a field of characteristic $0$ and $k$ is a positive integer. When $ξ$ is specialized to a positive integer $n$, we establish the Schur-Weyl duality between the actions of resulting algebra $\mathcal{MP}_k(n)$ and the symmetric group $S_n$ on $\text{Sym}^k(F^n)$. The construction of $\mathcal{MP}_k(ξ)$ generalizes to any vector $λ$ of non-negative integers yielding the algebra $\mathcal{MP}_λ(ξ)$ over $F[ξ]$ so that there is Schur-Weyl duality between the actions of $\mathcal{MP}_λ(n)$ and $S_n$ on $\text{Sym}^λ(F^n)$. We find the generating function for the multiplicity of each irreducible representation of $S_n$ in $\text{Sym}^λ(F^n)$, as $λ$ varies, in terms of a plethysm of Schur functions. As consequences we obtain an indexing set for the irreducible representations of $\mathcal{MP}_k(n)$, and the generating function for the multiplicity of an irreducible polynomial representation of $GL_n(F)$ when restricted to $S_n$. We show that $\mathcal{MP}_λ(ξ)$ embeds inside the partition algebra $\mathcal{P}_{|λ|}(ξ)$. Using this embedding, over $F$, we prove that $\mathcal{MP}_λ(ξ)$ is a cellular algebra, and $\mathcal{MP}_λ(ξ)$ is semisimple when $ξ$ is not an integer or $ξ$ is an integer such that $ξ\geq 2|λ|-1$. We give an insertion algorithm based on Robinson-Schensted-Knuth correspondence realizing the decomposition of $\mathcal{MP}_λ(n)$ as $\mathcal{MP}_λ(n)\times \mathcal{MP}_λ(n)$-module.
We consider a tower of generalized rook monoid algebras over the field $\mathbb{C}$ of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys-Murphy elements for generalized rook monoid algebras. Over an algebraically closed field $\Bbbk$ of positive characteristic $p$, utilizing Jucys-Murphy elements of rook monoid algebras, for $0\leq i\leq p-1$ we define the corresponding $i$-restriction and $i$-induction functors along with two extra functors. On the direct sum $\mathcal{G}_{\mathbb{C}}$ of the Grothendieck groups of module categories over rook monoid algebras over $\Bbbk$, these functors induce an action of the tensor product of the universal enveloping algebra $U(\hat{\mathfrak{sl}}_p(\mathbb{C}))$ and the monoid algebra $\mathbb{C}[\mathcal{B}]$ of the bicyclic monoid $\mathcal{B}$. Furthermore, we prove that $\mathcal{G}_{\mathbb{C}}$ is isomorphic to the tensor product of the basic representation of $U(\hat{\mathfrak{sl}}_{p}(\mathbb{C}))$ and the unique infinite-dimensional simple module over $\mathbb{C}[\mathcal{B}]$, and also exhibit that $\mathcal{G}_{\mathbb{C}}$ is a bialgebra. Under some natural restrictions on the characteristic of $\Bbbk$, we outline the corresponding result for generalized rook monoids.
We introduce and study a multiparameter colored partition category $\mathcal{CPar}(\textbf{x})$ by extending the construction of the partition category, over an algebraically closed field $\Bbbk$ of characteristic zero and for a multiparameter $\textbf{x}\in \Bbbk^{r}$. The morphism spaces in $\mathcal{CPar}(\textbf{x})$ have bases in terms of partition diagrams whose parts are colored by elements of the multiplicative cyclic group $C_r$. We show that the endomorphism spaces of $\mathcal{CPar}(\textbf{x})$ and additive Karoubi envelope of $\mathcal{CPar}(\textbf{x})$ are generically semisimple. The category $\mathcal{CPar}(\textbf{x})$ is rigid symmetric strict monoidal and we give a presentation of $\mathcal{CPar}(\textbf{x})$ as a monoidal category. The path algebra of $\mathcal{CPar}(\textbf{x})$ admits a triangular decomposition with Cartan subalgebra being equal to the direct sum of the group algebras of complex reflection groups $G(r,n)$. We compute the structure constants for the classes of simple modules in the split Grothendieck ring of the category of modules over the path algebra of the downward partition subcategory of $\mathcal{CPar}(\textbf{x})$ in two ways. Among other things, this gives a closed formula for the product of the reduced Kronecker coefficients in terms of the Littlewood--Richardson coefficients for $G(r,n)$ and certain Kronecker coefficients for the wreath product $(C_r \times C_r)\wr S_n$. For $r=1$, this formula reduces to a formula for the reduced Kronecker coefficients given by Littlewood. We also give two analogues of the Robinson--Schensted correspondence for colored partition diagrams and, as an application, we classify the equivalence classes of Green's left, right and two-sided relations for the colored partition monoid in terms of these correspondences.
We study various aspects of the structure and representation theory of singular Artin monoids. This includes a number of generalizations of the desingularization map and explicit presentations for certain finite quotient monoids of diagrammatic nature. The main result is a categorification of the classical desingularization map for singular Artin monoids associated to finite Weyl groups using BGG category $\mathcal{O}$.
Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.
Kudryavtseva and Mazorchuk exhibited Schur-Weyl duality between the rook monoid algebra $\mathbb{C}R_n$ and the subalgebra $\mathbb{C}I_k$ of the partition algebra $\mathbb{C} A_k(n)$ acting on $(\mathbb{C}^n)^{\otimes k}$. In this paper, we consider a subalgebra $\mathbb{C}I_{k+\frac{1}{2}}$ of $\mathbb{C} I_{k+1}$ such that there is Schur-Weyl duality between the actions of $\mathbb{C} R_{n-1}$ and $\mathbb{C} I_{k+\frac{1}{2}}$ on $(\mathbb{C}^n)^{\otimes k}$. This paper studies the representation theory of partition algebras $\mathbb{C}I_k$ and $\mathbb{C}I_{k+\frac{1}{2}}$ for rook monoids inductively by considering the multiplicity free tower $$\mathbb{C} I_1\subset \mathbb{C} I_{\frac{3}{2}}\subset \mathbb{C} I_{2}\subset \cdots\subset \mathbb{C} I_{k}\subset \mathbb{C} I_{k+\frac{1}{2}}\subset\cdots.$$ Furthermore, this inductive approach is established as a spectral approach by describing the Jucys-Murphy elements and their actions on the canonical Gelfand-Tsetlin bases, determined by the aforementioned multiplicity free tower, of irreducible representations of $\mathbb{C} I_k$ and $\mathbb{C} I_{k+\frac{1}{2}}$. Also, we describe the Jucys-Murphy elements of $\mathbb{C} R_n$ which play a central role in the demonstration of the actions of Jucys-Murphy elements of $\mathbb{C} I_k$ and $\mathbb{C}I_{k+\frac{1}{2}}$.
We construct the polynomial induction functor, which is the right adjoint to the restriction functor from the category of polynomial representations of a general linear group to the category of representations of its Weyl group. This construction leads to a representation-theoretic proof of Littlewood's plethystic formula for the multiplicity of an irreducible representation of the symmetric group in such a restriction. The unimodality of certain bipartite partition functions follows.
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 $d$.
This paper defines the partition algebra for complex reflection group $G(r,p,n)$ acting on $k$-fold tensor product $(\mathbb{C}^n)^{\otimes k}$, where $\mathbb{C}^n$ is the reflection representation of $G(r,p,n)$. A basis of the centralizer algebra of this action of $G(r,p,n)$ was given by Tanabe and for $p =1$, the corresponding partition algebra was studied by Orellana. We also establish a subalgebra as partition algebra of a subgroup of $G(r,p,n)$ acting on $(\mathbb{C}^n)^{\otimes k}$. We call these algebras as Tanabe algebras. The aim of this paper is to study representation theory of Tanabe algebras: parametrization of their irreducible modules, and construction of Bratteli diagram for the tower of Tanabe algebras. We conclude the paper by giving Jucys-Murphy elements of Tanabe algebras and their actions on the Gelfand-Tsetlin basis, determined by this multiplicity free tower, of irreducible modules.
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 $k$ 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.
We study subfield-subcodes of Generalized Toric (GT) codes over $\mathbb{F}_{p^s}$. These are the multidimensional analogues of BCH codes, which may be seen as subfield-subcodes of generalized Reed-Solomon codes. We identify polynomial generators for subfield-subcodes of GT codes which allows us to determine the dimensions and obtain bounds for the minimum distance. We give several examples of binary and ternary subfield-subcodes of GT codes that are the best known codes of a given dimension and length.