We present new, simple proofs for the enumeration of five of the ten symmetry classes of plane partitions contained in a given box. Four of them are derived from a simple determinant evaluation, using combinatorial arguments. The previous proofs of these four cases were quite complicated. For one more symmetry class we give an elementary proof in the case when two of the sides of the box are equal. Our results include simple evaluations of the determinants $\det\big(\delta_{ij}+\binom{x+i+j}{i}\big)_{0\le i,\, j\le n-1}$ and $\det\big(\binom{x+i+j}{2j-i}\big)_{0\le i,\,j \le n-1}$, notorious in plane partition enumeration, whose previous evaluations were quite intricate.

We consider a category of finite crystals of a quantum affine algebra whose objects are not necessarily perfect, and set of paths, semiinfinite tensor product of an object of this category with a certain boundary condition. It is shown that the set of paths is isomorphic to a direct sum of infinitely many, in general, crystals of integrable highest weight modules. We present examples from $C_n^{(1)}$ and $A_{n-1}^{(1)}$, in which the direct sum becomes a tensor product as suggested from the Bethe Ansatz.

In this paper we treat the minor summation formulas of pfaffians presented in [IW1] and derive several basic formulas concerning pfaffians from it. We also present a pfaffian version of the Plücker relation and give a new pfaffian identity as its application.

We show that the Kazhdan–Lusztig basis elements $C_w$ of the Hecke algebra of the symmetric group, when $w \in S_n$ corresponds to a Schubert subvariety of a Grassmann variety, can be written as a product of factors of the form $T_i + f_j(v)$, where $f_j$ are rational functions.

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 introduced in [21] in terms of ribbon tableaux.

A weight basis for each finite-dimensional irreducible representation of the orthogonal Lie algebra $\mathfrak{o}(2n)$ is constructed. The basis vectors are parametrized by the $D$-type Gelfand–Tsetlin patterns. The basis is consistent with the chain of subalgebras ${\mathfrak{g}}_1 \subset \cdots \subset {\mathfrak{g}}_n$, where ${\mathfrak{g}}_k = \mathfrak{o}(2k)$. Explicit formulas for the matrix elements of generators of $\mathfrak{o}(2n)$ in this basis are given. The construction is based on the representation theory of the Yangians and extends our previous results for the symplectic Lie algebras.

Weight vectors of the basic representations of $A^{(2)}_{2\ell}$ and $D^{(2)}_{\ell+1}$ are studied. They are expressed in terms of Schur’s $Q$-functions. The up and down motion along the string of the fundamental imaginary root is described as a combinatorial game.

Irreducible characters of Hecke algebras of type $A$ may be represented as refined counts of simple statistics on suitable subsets of permutations. Such formulas have been generalized to characters of other Coxeter groups and their Hecke algebras and to coinvariant algebras. In this paper we present several formulas, applications to combinatorial identities, and related problems. New results are given with proofs.

We prove a bosonic formula for the generating function of levelrestricted paths for the nonexceptional affine Kac–Moody algebras. In affine type A this yields an expression for the level-restricted generalized Kostka polynomials.

In this paper, we construct a length function $n(w)$ for the complex reflection group $W = G(r, p, n)$ by making use of certain partitions of the root system associated to $\widetilde{W} = G(r, 1, n)$. We show that the function $n(w)$ yields the Poincaré polynomial $P_W(q)$. We give some characterization of this function in a way independent of the choice of the root system.

We study the representation theory of the degenerate affine Hecke algebra $H_\ell$ of $GL_\ell$ using functors that connect the representation theory of $H_\ell$ and that of the Lie algebra $\mathfrak{gl}_n$. In particular, a new algebraic approach to the classification theorem of simple $H_\ell$-modules is given.

In this article, we give a recursion formula of the weighted parabolic Kazhdan–Lusztig polynomials and describe a relationship between those polynomials and weighted Kazhdan–Lusztig polynomials introduced by G.Lusztig ([4]).

We will review recent developments on the special polynomials arising in Painlevé equations and their generalizations.