Related papers: Universal factorial Schur $P,Q$-functions and thei…

In this paper, we study the generalized (co)homology Hopf algebras of the loop spaces on the infinite classical groups, generalizing the work due to Kono-Kozima and Clarke. We shall give a description of these Hopf algebras in terms of…

We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies…

We find presentations by generators and relations for the equivariant quantum cohomology rings of the maximal isotropic Grassmannians of types B,C and D, and we find polynomial representatives for the Schubert classes in these rings. These…

Let $k$ be a commutative $\mathbb{Q}$-algebra. We study families of functors between categories of finitely generated $R$-modules which are defined for all commutative $k$-algebras $R$ simultaneously and are compatible with base changes.…

We introduce two families of symmetric functions generalizing the factorial Schur $P$- and $Q$- functions due to Ivanov. We call them $K$-theoretic analogues of factorial Schur $P$- and $Q$- functions. We prove various combinatorial…

In this paper, we introduce {\it factorial} analogues of the ordinary Hall--Littlewood $P$- and $Q$-polynomials, which we call the {\it factorial Hall--Littlewood $P$- and $Q$-polynomials}. Using the {\it universal} formal group law, we…

We introduce and study a generalization of Schur's $P$-/$Q$-functions associated to a polynomial sequence, which can be viewed as ``Macdonald's ninth variation'' for $P$-/$Q$-functions. This variation includes as special cases Schur's…

We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $\operatorname{GL}_n$, the two-parameter…

The Schur class, denoted by $\mathcal{S}(\mathbb{D})$, is the set of all functions analytic and bounded by one in modulus in the open unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$, that is \[ \mathcal{S}(\mathbb{D}) = \{\varphi…

The Schubert bases of the torus-equivariant homology and cohomology rings of the affine Grassmannian of the special linear group are realized by new families of symmetric functions called k-double Schur functions and affine double Schur…

We introduce the quantum multi-Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations the quantum double Schubert polynomial coincides with some quantum…

We use category theory to propose a unified approach to the Schur-Weyl dualities involving the general linear Lie algebras, their polynomial extensions and associated quantum deformations. We define multiplicative sequences of algebras…

A general theory of matrix-spherical functions for dual Hopf algebras and right coideal subalgebras is developed. We establish their existence and define their orthogonality relations. When specialized to Kolb and Letzter's quantum…

A factorial analogue of the supersymmetric Schur functions is introduced. It is shown that factorial versions of the Jacobi--Trudi and Sergeev--Pragacz formulae hold. The results are applied to construct a linear basis in the center of the…

Consider the algebra Q<<x_1,x_2,...>> of formal power series in countably many noncommuting variables over the rationals. The subalgebra Pi(x_1,x_2,...) of symmetric functions in noncommuting variables consists of all elements invariant…

For each infinite series of the classical Lie groups of type B,C or D, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in…

We introduce the multiple zeta functions with structures similar to those of symmetric functions such as Schur $P$-, Schur $Q$-, symplectic and orthogonal functions in the representation theory. We first consider their basic properties such…

We give formulas for the products of classes of Schubert varieties in the quantum cohomology rings of Grassmannians, in terms of the combinatorics of partitions and tableaux.

We prove the Cauchy type identities for the universal double Schubert polynomials, introduced recently by W. Fulton. As a corollary, the determinantal formulae for some specializations of the universal double Schubert polynomials…

We construct a category of quantum polynomial functors which deforms Friedlander and Suslin's category of strict polynomial functors. The main aim of this paper is to develop from first principles the basic structural properties of this…