The integral shuffle algebra and the $K$-theory of the Hilbert scheme of points in $\mathbb{A}^2$

Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra

In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C^2. We show that commutative elements K_i of shuffle algebra act through vertex operators over positive part {h_i}_{i>0} of the Heisenberg algebra in these K-groups. Hence we get the action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space k[h_1,h_2,...].

On Poincare polynomials of shuffle algebra representations

In this paper we study some combinatorial properties of the shuffle algebra $S_{q_{1},q_{2}}$ with two complex parameters and, in particular, obtain the Hilbert series for $S_{q_{1},q_{2}}/I$, where $I$ is the ideal generated by $z-λ$ for both generic $q_{1},q_{2}$ and satisfying the relation $q_{1}^{a}q_{2}^{b}=1$ with $a, b \in \mathbb{N}$.

PBWD bases and shuffle algebra realizations for $U_v(L\mathfrak{sl}_n), U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(m|n))$ and their integral forms

We construct a family of PBWD (Poincaré-Birkhoff-Witt-Drinfeld) bases for the quantum loop algebras $U_v(L\mathfrak{sl}_n), U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(m|n))$ in the new Drinfeld realizations. In the 2-parameter case, this proves Theorem 3.11 of [Hu-Rosso-Zhang] (stated in loc. cit. without a proof), while in the super case it proves a conjecture of [Zhang]. The main ingredient in our proofs is the interplay between those quantum loop algebras and the corresponding shuffle algebras, which are trigonometric counterparts of the elliptic shuffle algebras of Feigin-Odesskii. …

洗牌代数及其整形式：Cₙ和Dₙ型中的特殊化映射方法

Shuffle algebras and their integral forms: specialization map approach in types $C_n$ and $D_n$

针对Cₙ和Dₙ型量子环代数正子代数及其整形式，通过新Drinfeld实现构造PBWD基，建立洗牌代数实现并推广到整形式，自然推广经典型处理且特殊化映射更优。

Shuffle algebras, lattice paths and the commuting scheme

The commutative trigonometric shuffle algebra ${\mathrm A}$ is a space of symmetric rational functions satisfying certain wheel conditions. We describe a ring isomorphism between ${\mathrm A}$ and the center of the Hecke algebra using a realization of the elements of ${\mathrm A}$ as partition functions of coloured lattice paths associated to the $R$-matrix of $\mathcal U_{t^{1/2}}(\widehat{gl}_{\infty})$. As an application, we compute under certain conditions the Hilbert series of the commuting scheme and identify it with a particular element of the shuffle algebra ${\mathrm A}$, thus providi…

Shuffle 代数及其整形式：$B_n$ 型与 $G_2$ 型中的特化映射方法

Shuffle algebras and their integral forms: specialization map approach in types $B_n$ and $G_2$

本文利用特化映射方法，为 $B_n$ 与 $G_2$ 型量子环面代数正子代数及其整形式构造 PBWD 基，并建立相应 shuffle 代数实现，推广了 $A_n$ 型结果。