Deformed universal characters for classical and affine algebras

Deformed Macdonald-Ruijsenaars operators and super Macdonald polynomials

It is shown that the deformed Macdonald-Ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual Macdonald-Ruijsenaars operator in infinite number of variables. The ideals of these varieties are shown to be generated by the Macdonald polynomials related to Young diagrams with special geometry. The super Macdonald polynomials and their shifted version are introduced, the combinatorial formulas for them are given.

Whittaker vector of deformed Virasoro algebra and Macdonald symmetric functions

We give a proof of Awata and Yamada's conjecture for the explicit formula of Whittaker vector of the deformed Virasoro algebra realized in the Fock space. The formula is expressed as a summation over Macdonald symmetric functions with factored coefficients. In the proof we fully use currents appearing in the Fock representation of Ding-Iohara-Miki quantum algebra. We also mention an interpretation of Whittaker vector in terms of the geometry of the Hilbert schemes of points on the affine plane.

Multipartitions, Generalized Durfee Squares and Affine Lie Algebra Characters

We give some higher dimensional analogues of the Durfee square formula and point out their relation to dissections of multipartitions. We apply the results to write certain affine Lie algebra characters in terms of Universal Chiral Partition Functions.

q-deformed Character Theory for Infinite-Dimensional Symplectic and Orthogonal Groups

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding finite-dimensional groups, as the rank tends to infinity. We solve a q-deformed version of the latter problem for orthogonal and symplectic groups, extending previously known results for the unitary group. The proof is based on novel determinantal and double-contour integral formulas for the q-specialized characters.

仿射格拉斯曼流形的等变 $K$-同调与 $K$-理论双重 $k$-Schur 函数

Equivariant $K$-homology of affine Grassmannian and $K$-theoretic double $k$-Schur functions

本文研究仿射格拉斯曼流形的环面等变 $K$-同调环，在 $A$ 型中引入 $K$-理论双重 $k$-Schur 函数作为 Schubert 基，并给出 $\mathrm{Gr}_{{SL}_n}$ 的等变 $K$-同调环的 Ginzburg-Peterson 型实现。

Combinatorics of vertex operators and deformed $W$-algebra of type D$(2,1;α)$

We consider sets of screening operators with fermionic screening currents. We study sums of vertex operators which formally commute with the screening operators assuming that each vertex operator has rational contractions with all screening currents with only simple poles. We develop and use the method of $qq$-characters which are combinatorial objects described in terms of deformed Cartan matrix. We show that each qq-character gives rise to a sum of vertex operators commuting with screening operators and describe ways to understand the sum in the case it is infinite. We discuss combinatorics …