相关论文: Symmetric Polynomials and $U_q(\hat{sl}_2)$

We introduce a new family of symmetric functions, which are $q$-analogues of products of Schur functions defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation of the quantum affine…

We introduce and study several combinatorial properties of a class of symmetric polynomials from the point of view of integrable vertex models in finite lattice. We introduce the $L$-operator related with the $U_q(sl_2)$ $R$-matrix, and…

We categorify Lusztig's version of the quantized enveloping algebra for sl(2). Using a graphical calculus a 2-category is constructed whose split Grothendieck ring is isomorphic to Lusztig's algebra. The indecomposable morphisms of this…

We establish a ring isomorphism between the derived Hall algebra of the Jordan quiver and the ring of double symmetric functions (i.e., the ring of symmetric polynomials in two sets of countably many variables, invariant under the…

It is well known that if one integrates a Schur function indexed by a partition $\lambda$ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of $\lambda$ have even multiplicity (resp. all parts of $\lambda$…

We consider the expansion of the square of a complete homogeneous function $h_\lambda$, or of an elementary symmetric function $e_\lambda$, in the basis of Schur functions. This square also decomposes into two plethysms, $s_2[h_\lambda]$…

We study some classes of symmetric operators for the discrete series representations of the quantum algebra U_q(su_{1,1}), which may serve as Hamiltonians of various physical systems. The problem of diagonalization of these operators…

The $q$-Whittaker function $W_\lambda(\mathbf{x};q)$ associated to a partition $\lambda$ is a $q$-analogue of the Schur function $s_\lambda(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial…

We show how to use Clifford algebra techniques to describe the de Rham cohomology ring of equal rank compact symmetric spaces $G/K$. In particular, for $G/K=U(n)/U(k)\times U(n-k)$, we obtain a new way of multiplying Schur polynomials,…

We obtain a new formula to relate the value of a Schur polynomial with variables $(x_1,\ldots,x_N)$ with values of Schur polynomials at $(1,\ldots,1)$. This allows to study the limit shape of perfect matchings on a square hexagon lattice…

Let $H$ be the Iwahori-Hecke algebra and let $J$ be Lusztig's asymptotic Hecke algebra, both specialized to type $\tilde{A}_1$. For $\mathrm{SL}_2$, when the parameter $q$ is specialized to a prime power, Braverman and Kazhdan showed…

We study three-dimensional partition functions constructed from the tetrahedral $L$-operator introduced and studied by Bazhanov-Sergeev and Kuniba-Maruyama-Okado. First, we explore the $q=0$ case, extending the authors' previous results and…

We show that the Littlewood-Richardson coefficients are values at 1 of certain parabolic Kazhdan-Lusztig polynomials for affine symmetric groups. These q-analogues of Littlewood-Richardson multiplicities coincide with those previously…

The double Schur functions form a distinguished basis of the ring \Lambda(x||a) which is a multiparameter generalization of the ring of symmetric functions \Lambda(x). The canonical comultiplication on \Lambda(x) is extended to…

We present a library of formalized results around symmetric functions and the character theory of symmetric groups. Written in Coq/Rocq and based on the Mathematical Components library, it covers a large part of the contents of a graduate…

A class of second order difference (discrete) operators with a partial algebraization of the spectrum is introduced. The eigenfuncions of the algebraized part of the spectrum are polinomials (discrete polinomials). Such difference operators…

We introduce and study a one-parameter generalization of the q-Whittaker symmetric functions. This is a family of multivariate symmetric polynomials, whose construction may be viewed as an application of the procedure of fusion from…

We present a family of analogs of the Hall-Littlewood symmetric functions in the $Q$-function algebra. The change of basis coefficients between this family and Schur's $Q$-functions are $q$-analogs of numbers of marked shifted tableaux.…

Notes from a course at the ATM Workshop on Schubert Varieties, held at The Institute of Mathematical Sciences, Chennai, in November 2017. Various expansions of Schur functions, the Lindstr\"om-Gessel-Viennot lemma, semistandard Young…

In this paper we study higher level Deligne--Lusztig representations of reductive groups over discrete valuation rings, with finite residue field $\mathbb{F}_q$. In previous work we proved that, at even levels, these geometrically…