Related papers: Free-fermions and skew stable Grothendieck polynom…

Yeliussizov has classified the positive specializations of symmetric Grothendieck functions, defined in several different ways, providing a K-theoretic lift of the classical Edrei-Thoma theorem. This note studies the analogous…

LLT polynomials are $q$-analogues of product of Schur functions that are known to be Schur-positive by Grojnowski and Haiman. However, there is no known combinatorial formula for the coefficients in the Schur expansion. Finding such a…

The present paper is a detailed version of math/0003031. We introduce and study a new basis in the algebra of symmetric functions. The elements of this basis are called the Frobenius-Schur functions (FS-functions, for short). Our main…

The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. An analogous statement is proved,…

We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable categories have a structure of triangulated categories. Our main result determines the…

Let $X^N$ be a family of $N\times N$ independent GUE random matrices, $Z^N$ a family of deterministic matrices, $P$ a self-adjoint non-commutative polynomial, that is for any $N$, $P(X^N)$ is self-adjoint, $f$ a smooth function. We prove…

Let $\Lambda$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_\lambda[X/(1-t)]\}_{\lambda_1\leq k}$. We introduce a new family of symmetric polynomials,…

We prove that the combinatorial side of the "Rational Shuffle Conjecture" provides a Schur-positive symmetric polynomial. Furthermore, we prove that the contribution of a given rational Dyck path can be computed as a certain skew LLT…

We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for…

The classical theory of free analysis generalizes the noncommutative (nc) polynomials and rational functions, easily providing such results as an nc analogue of the Jacobian conjecture. However, the classical theory misses out on important…

The ring of symmetric functions $\Lambda$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the…

We use mirror formulas for the stable quotients analogue of Givental's J-function for twisted projective invariants obtained in a previous paper to obtain mirror formulas for the analogues of the double and triple Givental's J-functions…

We introduce and investigate a family of tau-functions of the 2D Toda hierarchy, which is a natural generalization of the hypergeometric family associated with Hurwitz numbers. For this family we prove a skew Schur function expansion…

In this paper we classify when (row-strict) dual immaculate functions and (row-strict) extended Schur functions, as well as their skew generalizations, are symmetric. We also classify when their natural variants, termed advanced functions,…

We obtain generalized Christoffel-Darboux (GCD) formula for skew-orthogonal polynomials (SOP). Using this, we present an alternative derivation of the level density and two-point function for Gaussian orthogonal ensembles (GOE) and Gaussian…

The paper extends the classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide the uniform asymptotic expansion in terms of the Hermite polynomials. We explicitly obtain expressions for a few…

We prove that twisted versions of Schubert polynomials defined by $\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$ and $\widetilde{\mathfrak S}_{ws_i} = (s_i+\partial_i)\widetilde{\mathfrak S}_w$ are monomial positive and…

In this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let $S$ be a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree $1$ and $f =x_1^2 + \cdots +x_n^2…

The machinery of noncommutative Schur functions provides a general tool for obtaining Schur expansions for combinatorially defined symmetric functions. We extend this approach to a wider class of symmetric functions, explore its strengths…

We create several families of bases for the symmetric polynomials. From these bases we prove that certain Schur symmetric polynomials form a basis for quotients of symmetric polynomials that generalize the cohomology and the quantum…