A multiplicative property of quantum flag minors II

On dual canonical bases

The dual basis of the canonical basis of the modified quantized enveloping algebra is studied, in particular for type $A$. The construction of a basis for the coordinate algebra of the $n\times n$ quantum matrices is appropriate for the study the multiplicative property. It is shown that this basis is invariant under multiplication by certain quantum minors including the quantum determinant. Then a basis of quantum SL(n) is obtained by setting the quantum determinant to one. This basis turns out to be equivalent to the dual canonical basis.

A geometric realisation of 0-Schur and 0-Hecke algebras

We define a new product on orbits of pairs of flags in a vector space, using open orbits in certain varieties of pairs of flags. This new product defines an associative $\mathbb{Z}$-algebra, denoted by $G(n,r)$. We show that $G(n,r)$ is a geometric realisation of the 0-Schur algebra $S_0(n, r)$ over $\mathbb{Z}$, which is the $q$-Schur algebra $S_q(n,r)$ at q=0. We view a pair of flags as a pair of projective resolutions for a quiver of type $\mathbb{A}$ with linear orientation, and study $q$-Schur algebras from this point of view. This allows us to understand the relation between $q$-Schur al…

Quantum Unipotent Subgroup and dual canonical basis

Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate ring $C[N(w)]$ of the unipotent subgroup, associated with a Weyl group element $w$ and they proved cluster monomials are contained in Lusztig's dual semicanonical basis $S^{*}$. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras to the dual canonical basis ${B}^{up}$. In particular, we prove that the quantum analogue $O_{q}[N(w)]$ of ${C}[N(w)]$ has the induced basis from ${B}^{up}$, which contains quantum flag minors and satisfies a factorizatio…

有限群轨道相关的熵

Entropies associated with orbits of finite groups

本文研究有限反射群和辛群轨道的熵，通过信息论视角揭示新熵函数及其与传统熵的关系。

Equivariant multiplicities via representations of quantum affine algebras

For any simply-laced type simple Lie algebra $\mathfrak{g}$ and any height function $ξ$ adapted to an orientation $Q$ of the Dynkin diagram of $\mathfrak{g}$, Hernandez-Leclerc introduced a certain category $\mathcal{C}^{\leq ξ}$ of representations of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$, as well as a subcategory $\mathcal{C}_Q$ of $\mathcal{C}^{\leq ξ}$ whose complexified Grothendieck ring is isomorphic to the coordinate ring $\mathbb{C}[\mathbf{N}]$ of a maximal unipotent subgroup. In this paper, we define an algebraic morphism $\widetilde{D}_ξ$ on a torus $\mathcal{Y}^{\…