Related papers: $\imath$Schur duality and Kazhdan-Lusztig basis ex…

We formulate a $q$-Schur algebra associated to an arbitrary $W$-invariant finite set $X_{\texttt f}$ of integral weights for a complex simple Lie algebra with Weyl group $W$. We establish a $q$-Schur duality between the $q$-Schur algebra…

The generic Hecke algebra for the hyperoctahedral group, i.e. the Weyl group of type B, contains the generic Hecke algebra for the symmetric group, i.e. the Weyl group of type A, as a subalgebra. Inducing the index representation of the…

We extend Schur-Weyl duality to an arbitrary level $l \geq 1$, the case $l=1$ recovering the classical duality between the symmetric and general linear groups. In general, the symmetric group is replaced by the degenerate cyclotomic Hecke…

In 2012 Raghavan, Samuel, and Subrahmanyam showed that the Kazhdan--Lusztig basis for the Iwahori--Hecke algebra in type A provides a ``canonical'' basis for the centraliser algebra of the Schur algebra acting on tensor space. In 2022 the…

We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional…

We generalize a construction in [BW18] (arXiv:1610.09271) by showing that the tensor product of a based $\textbf{U}^{\imath}$-module and a based $\textbf{U}$-module is a based $\textbf{U}^{\imath}$-module. This is then used to formulate a…

As a natural generalization quantum Schur algebras associated with the Hecke algebra of the symmetric group, we introduce the quantum Schur superalgebra of type Q associated with the Hecke-Clifford superalgebra, which, by definition, is the…

In this paper we show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the group $S_n$ coincide with the coefficients of the canonical basis in $n$th tensor power of the fundamental representation of…

We introduce the notion of quantum Schur (or $q$-Schur) superalgebras. These algebras share certain nice properties with $q$-Schur algebras such as base change property, existence of canonical $\mathbb Z[v,v^{-1}]$-bases, and the duality…

Let $G$ be a finite group of Lie type. In studying the cross-characteristic representation theory of $G$, the (specialized) Hecke algebra $H=\End_G(\ind_B^G1_B)$ has played a important role. In particular, when $G=GL_n(\mathbb F_q)$ is a…

In a recent paper with Sahi and Stokman, we introduced quasi-polynomial generalizations of Macdonald polynomials for arbitrary root systems via a new class of representations of the double affine Hecke algebra. These objects depend on a…

We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases…

Dual canonical bases of the quantum general linear supergroup are constructed which are invariant under the multiplication of the quantum Berezinian. By setting the quantum Berezinian to identity, we obtain dual canonical bases of the…

We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasiLascoux basis, which is simultaneously both a $K$-theoretic deformation of the quasikey basis and also a lift of the…

From a category $\mathcal{A}$ with an involution $\varrho$, we introduce $\varrho$-complexes, which are a generalization of (bounded) complexes, periodic complexes and modules of $\imath$quiver algebras. The homological properties of the…

We develop algebraic and geometrical approaches toward canonical bases for affine q-Schur algebras of arbitrary type introduced in this paper. A duality between an affine q-Schur algebra and a corresponding affine Hecke algebra is…

We consider the invariants Ker and Im for commutative squares in quasi-abelian categories. These invariants were introduced by Lambek for groups and then studied by Hilton and Nomura in exact categories.

We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space Q_k of quasi-invariants of a given multiplicity is not, in general, an algebra but a module over the…

A quantum symmetric pair consists of a quantum group $\mathbf U$ and its coideal subalgebra ${\mathbf U}^{\imath}_{\boldsymbol{\varsigma}}$ with parameters $\boldsymbol{\varsigma}$ (called an $\imath$quantum group). We initiate a Hall…

Let $(\bf U, \bf U^\imath)$ be a quasi-split quantum symmetric pair of arbitrary Kac-Moody type, where "quasi-split" means the corresponding Satake diagram contains no black node. We give a presentation of the $\imath$quantum group $\bf…