Comet-shaped quiver varieties, Weyl group actions, and modified Kostka polynomials

On cyclic quiver parabolic Kostka-Shoji polynomials

We obtain an explicit combinatorial formula for certain parabolic Kostka-Shoji polynomials associated with the cyclic quiver, generalizing results of Shoji and of Liu and Shoji.

Quiver Hall-Littlewood functions and Kostka-Shoji polynomials

For any triple $(i,a,μ)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $μ$, we define a quiver current $H^{(i,a)}_μ$ acting on the tensor power $Λ^Q$ of symmetric functions over the vertices of $Q$. These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple $(\mathbf{i},\mathbf{a},μ(\bullet))$ of sequences of such data, we define the quiver Hall-Littlewood function $H^{\mathbf{i},\mathbf{a}}_{μ(\bullet)}$ as the result of acting on $1\inΛ^Q$ by the corres…

Kostka-Shoji polynomials and Lusztig's convolution diagram

We propose an $r$-variable version of Kostka-Shoji polynomials $K^-_{λμ}$ for $r$-multipartitions $λ,μ$. Our version has positive integral coefficients and encodes the graded multiplicities in the space of global sections of a line bundle over Lusztig's iterated convolution diagram for the cyclic quiver $\tilde{A}_{r-1}$.

An Explicit Description of the B(\infty) Crystal For Generalized Quantum Groups of a Family of Comet Quivers

Tristan Bozec gave a definition of generalized quantum groups that extends the usual definition of quantum groups to finite quivers with loops at vertices, and he introduced a theory of generalized crystals for this new family of Hopf algebras. We explicitly characterize the generalized crystal $\mathcal{B}(\infty)$ associated to a certain family of quivers with multiple loops by providing a complete set of relations among the Kashiwara operators themselves.

关于King-Tollu-Toumazet矩形Kostka系数的一个猜想

On a conjecture of King-Tollu-Toumazet for rectangular Kostka coefficients

针对King-Tollu-Toumazet关于矩形Kostka系数的猜想，本文证明了该猜想对所有矩形Kostka系数成立，将其实现为Grassmannian齐次坐标环的重数等，通过稳定分解确定了猜想中的两种情形。

Arithmetic harmonic analysis on character and quiver varieties

We present a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a Riemann surface of genus g to GL_n(C) with fixed generic semi-simple conjugacy classes at k punctures. Using the character table of GL_n(F_q) we calculate the E-polynomial of these character varieties and confirm that it is as predicted by our main conjecture. Then, using the character table of gl_n(F_q), we calculate the E-polynomial of certain associated comet-shaped quiver varieties,…