Related papers: Positivity of Canonical Bases under Comultiplicati…
We provide examples for negativity of structure constants of the stably canonical basis of modified quantum $\mathfrak{gl}_n$ and an analogous basis of modified quantum coideal algebra of $\mathfrak{gl}_n$. In contrast, we construct the…
The positivity property for canonical bases asserts that the structure constants of the multiplication for the canonical basis are in ${\mathbb N}[v,v^{-1}]$. Let $\mathbf U$ be the quantum group over ${\mathbb Q}(v)$ associated with a…
We give a geometric interpretation of the inner product on the modified quantum group of $\hat{\mathfrak{sl}}_n$. We also give some applications of this interpretation, including a positivity result for the inner product, and a new…
Let $\dot{\mathbf{U}}(\widehat{\frak{sl}}_n)$ be the modified quantum affine $\frak{sl}_n$ and let ${\bf U}(\widehat{\frak{sl}}_N)^+$ be the positive part of quantum affine $\frak{sl}_N$. Let $\dot{\mathbf{B}}(n)$ be the canonical basis of…
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…
The main motivation for the study of cluster algebras initiated in math.RT/0104151, math.RA/0208229 and math.RT/0305434 was to design an algebraic framework for understanding total positivity and canonical bases in semisimple algebraic…
We establish a multiplication formula for a tridiagonal standard basis element in the idempotented coideal subalgebras of quantum affine $\mathfrak{gl}_n$ arising from the geometry of affine partial flag varieties of type $C$. We apply this…
We define a canonical basis of the $q$-deformed Fock space representation of the affine Lie algebra $\glchap_n$. We conjecture that the entries of the transition matrix between this basis and the natural basis of the Fock space are…
We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and…
We define canonical bases of the higher-level q-deformed Fock space modules of the affine Lie algebra sl(n)^. This generalizes the result of Leclerc and Thibon for the case of level 1. We express the transition matrices between the…
In this paper, we study the structures of Schur algebra and Lusztig algebra associated to partial flag varieties of affine type D. We show that there is a subalgebra of Lusztig algebra and the quantum groups arising from this subalgebras…
This represents a talk given at the International Conference for Basic Science, July 2025. We review the theory of canonical bases of quantum groups and its relation with the theory of total positivity.
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
We construct a monomial basis of the positive part of the quantized enveloping algebra associated to a finite-dimensional simple Lie algebra. As an application we give a simple proof of the existence and uniqueness of the canonical basis of…
We consider tensor products of finite-dimensional representations of a coideal subalgebra in $U_{q}(\mathfrak{sl}_2)$. We present an explicit expression for the dual of the canonical bases through a diagrammatic presentation. We show that…
A string basis is constructed for each subalgebra of invariants of the function algebra on the quantum special linear group. By analyzing the string basis for a particular subalgebra of invariants, we obtain a ``canonical basis'' for every…
For any affine Weyl group, we introduce the pre-canonical bases. They are a set of bases $\{\mathbf{N}^i\}_{1\leq i \leq m+1} $ (where $m$ is the height of the highest root) of the spherical Hecke algebra that interpolates between the…
Consider a decomposition $\mathfrak{n} = \mathfrak{n}_1 \oplus \cdots \oplus \mathfrak{n}_r$ of the positive nilradical of a complex semisimple Lie algebra of rank $r$, where each $\mathfrak{n}_k$ is a module under an appropriate Levi…
We introduce a modified quantum enveloping algebra as well as a (modified) covering quantum algebra for the ortho-symplectic Lie superalgebra osp(1|2). Then we formulate and compute the corresponding canonical bases, and relate them to the…
Dual canonical bases of the quantum general linear supergroup are constructed which are invariant under the multiplication of the quantum Berezinian. By setting the quantum Berezinian to identity, we obtain dual canonical bases of the…