Related papers: Skew Howe duality and random rectangular Young tab…

We consider two partial orders on standard Young tableaux. The first one is induced from the weak right Bruhat order on symmetric group by Robinson-Schensted algorithm. The second one is induced from the order on Young diagrams by…

Take the degenerate affine Hecke algebra $H_{l+m}$ corresponding to the group $GL_{l+m}$ over a $p$-adic field. Consider the $H_{l+m}$-module $W$ induced from the tensor product of the evaluation modules over the algebras $H_l$ and $H_m$.…

We study the generalization of S-duality to non-commutative gauge theories. For rank one theories, we obtain the leading terms of the dual theory by Legendre transforming the Lagrangian of the non-commutative theory expressed in terms of a…

Let $\mathbb{S}$ denote the oscillatory module over the complex symplectic Lie algebra $\mathfrak{g}= \mathfrak{sp}(\mathbb{V}^{\mathbb{C}},\omega).$ Consider the $\mathfrak{g}$-module…

We present some results about the irreducible representations appearing in the exterior algebra $\Lambda \mathfrak{g}$, where $ \mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$. For Lie algebras of type $B$, $C$ or $D$ we prove that…

We give a fermionic formula for $R$-matrices of exterior powers of the vector representations of $U_q(\widehat{ \mathfrak{gl}}_N)$ and relate it to the dynamical Weyl group of Tarasov--Varchenko and Etingof--Varchenko, via a Howe…

Schur-Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog…

The aim of this paper is to explain how to get a complex of smooth representations out of the dual vector space to a smooth representation of a p-adic Lie group, in natural characteristic. The construction does not depend on any…

Geometric Langlands duality relates a representation of a simple Lie group $G^\vee$ to the cohomology of a certain moduli space associated with the dual group $G$. In this correspondence, a principal $SL_2$ subgroup of $G^\vee$ makes an…

Skew-symmetric forms possess unique capabilities. The properties of closed exterior and dual forms, namely, invariance, covariance, conjugacy and duality, either explicitly or implicitly appear in all invariant mathematical formalisms. This…

Let $\lambda: \tilde{G}\to G$ be the non-trivial double covering of the symplectic group $G=Sp(V,\omega)$ of the symplectic vector space $(V,\omega)$ by the metaplectic group $\tilde{G}=Mp(V,\omega).$ In this case, $\lambda$ is also a…

This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong…

Over fields of characteristic zero, there are well known construction of the irreducible representations and of irreducible modules, called Specht modules for the symmetric groups $S_{n}$ which are based on elegant combinatorial concepts…

In this paper, using crystal theory we prove the existence of a new family of irreducible components appearing in the tensor product of two irreducible integrable highest weight modules over symmetrizable Kac-Moody algebras motivated by the…

We derive new combinatorial identities which may be viewed as multivariate analogs of summation formulas for hypergeometric series. As in the previous paper [Re], we start with probability distributions on the space of the infinite Young…

Let $K\left\langle X \right\rangle$ denote the free associative algebra generated by a set $X = \{x_1, \dots, x_n\}$ over a field $K$ of characteristic $0$. Let $I_p$, for $p \geq 2$, denote the two-sided ideal in $K\left\langle X…

Making use of a Howe duality involving the infinite-dimensional Lie superalgebra $\hgltwo$ and the finite-dimensional group $GL_l$ we derive a character formula for a certain class of irreducible quasi-finite representations of $\hgltwo$ in…

We study the symplectic Howe duality using two new and independent combinatorial methods: via determinantal formulae on the one hand, and via (bi)crystals on the other hand. The first approach allows us to establish a generalised version…

The crystals for a finite-dimensional complex reductive Lie algebra $\mathfrak{g}$ encode the structure of its representations, yet can also reveal surprising new structure of their own. We study the cactus group $C_{\mathfrak{g}}$,…

The S-dual $(\mathbf G^\vee\curvearrowright\mathbf M^\vee)$ of the pair $(\mathbf G\curvearrowright\mathbf M)$ of a smooth affine algebraic symplectic manifold $\mathbf M$ with hamiltonian action of a complex reductive group $\mathbf G$ was…