[Paper Review] Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebra of gl(2)

Stephen Doty, Anthony Giaquinto|ArXiv.org|Nov 21, 2000

Algebraic structures and combinatorial models2 references3 citations

TL;DR

This paper presents a new generators-and-relations description of the quantum Schur algebra $S_v(2,d)$ as a quotient of the quantized universal enveloping algebra $\mathbf{U}_v(\mathfrak{gl}_2)$, using explicit relations involving $e$, $f$, and $K_1^\pm$ or $K_2^\pm$. The key contribution is a new ${\mathbb{Z}}[v,v^{-1}]$-basis for the integral quantum Schur algebra closely related to Lusztig’s basis of the integral form of $\mathbf{U}_v(\mathfrak{gl}_2)$, achieved via an idempotent basis for the degree-zero part.

ABSTRACT

We obtain a presentation of quantum Schur algebras (over the field Q(v)) by generators and relations. This presentation is compatible with the usual presentation of the quantized universal enveloping algebra of the Lie algebra gl(2). We also locate the ``integral'' form of the quantum Schur algebra within the presented algebra and show it has a basis which is closely related to Lusztig's basis of the integral form of the quantized enveloping algebra.

Motivation & Objective

• To provide a presentation of the quantum Schur algebra $S_v(2,d)$ by generators and relations compatible with the standard presentation of $\mathbf{U}_v(\mathfrak{gl}_2)$.
• To construct a new ${\mathbb{Z}}[v,v^{-1}]$-basis for the integral quantum Schur algebra $S_{\mathcal{A}}(2,d)$ that is closely related to Lusztig’s basis of the integral form of $\mathbf{U}_v(\mathfrak{gl}_2)$.
• To replace the PBW-type basis used in classical Schur algebras with an idempotent basis for the degree-zero part, enabling generalization to higher $n$.
• To establish that the quantum Schur algebra over ${\mathbb{Q}}(v)$ is isomorphic to a quotient of $\mathbf{U}_v(\mathfrak{gl}_2)$ by a two-sided ideal generated by the minimal polynomial of $K_1$ (or $K_2$) and the relation $K_1K_2 = v^d$.

Proposed method

• The paper uses the representation $\rho_d: \mathbf{U}_v(\mathfrak{gl}_2) \to \operatorname{End}(E^{\otimes d})$ to define $S_v(2,d)$ as the image, then identifies the kernel via explicit relations.
• It introduces the $v$-binomial coefficients $\begin{bmatrix}K_1\ b\end{bmatrix}$ and $\begin{bmatrix}K_2\ b\end{bmatrix}$ as key generators for the degree-zero part of the algebra.
• The degree-zero part is shown to have an idempotent basis $\{K_{b_1,b_2}\}$ with $b_1 + b_2 = d$, which is used to construct the new integral basis.
• The proof relies on triangularity of the change-of-basis matrix between $\begin{bmatrix}K_1\ b\end{bmatrix}$ and $K_{b_1,b_2}$, ensuring invertibility over ${\mathcal{A}} = {\mathbb{Z}}[v,v^{-1}]$.
• Lemmas are used to show that any monomial $e^{(a)}\begin{bmatrix}K_1\ b\end{bmatrix}f^{(c)}$ with $a+b+c > d$ can be rewritten as an ${\mathcal{A}}$-linear combination of elements with total degree $\leq d$, enabling finite spanning sets.
• The final basis is constructed by showing that the set $\{e^{(a)}\begin{bmatrix}K_1\ b\end{bmatrix}f^{(c)} \mid a+b+c \leq d\}$ spans $S_{\mathcal{A}}(2,d)$ and has the correct cardinality, hence forms a basis.

Experimental results

Research questions

• RQ1 How can the quantum Schur algebra $S_v(2,d)$ be presented as a quotient of $\mathbf{U}_v(\mathfrak{gl}_2)$ using generators and relations compatible with the standard presentation?
• RQ2 What is the structure of the integral quantum Schur algebra $S_{\mathcal{A}}(2,d)$ over ${\mathbb{Z}}[v,v^{-1}]$ and how does it relate to Lusztig’s basis of $\mathbf{U}_{\mathcal{A}}$?
• RQ3 Can the degree-zero part of $S_v(2,d)$ be described using an idempotent basis instead of a PBW-type basis, and what are the implications for generalization?
• RQ4 What is the precise form of the kernel of the representation $\rho_d: \mathbf{U}_v(\mathfrak{gl}_2) \to \operatorname{End}(E^{\otimes d})$?
• RQ5 How can the basis of $S_{\mathcal{A}}(2,d)$ be constructed explicitly using $v$-binomial coefficients and divided powers?

Key findings

• The quantum Schur algebra $S_v(2,d)$ over ${\mathbb{Q}}(v)$ is isomorphic to the algebra generated by $e$, $f$, and $K_1^\pm$ subject to the relations: $KK^{-1} = 1$, $KeK^{-1} = v^2 e$, $KfK^{-1} = v^{-2} f$, and the minimal polynomial $(K - v^d)(K - v^{d-2})\cdots(K - v^{-d}) = 0$.
• An equivalent presentation uses $K_1$ with the relation $({K_1} - 1)({K_1} - v)\cdots({K_1} - v^d) = 0$, and the quantum Serre relation $ef - fe = \frac{v^{-d}K_1^2 - v^d K_1^{-2}}{v - v^{-1}}$.
• The integral quantum Schur algebra $S_{\mathcal{A}}(2,d)$ has an ${\mathcal{A}}$-basis $\{e^{(a)}\begin{bmatrix}K_1\ b\end{bmatrix}f^{(c)} \mid a + b + c \leq d\}$, which is closely related to Lusztig’s basis of $\mathbf{U}_{\mathcal{A}}$.
• The set $\{1, \begin{bmatrix}K_1\ 1\end{bmatrix}, \dots, \begin{bmatrix}K_1\ d\end{bmatrix}\}$ forms an ${\mathcal{A}}$-basis for the degree-zero part $S_{\mathcal{A}}^0(2,d)$, with the change-of-basis matrix being triangular with 1s on the diagonal.
• Any monomial $e^{(a)}\begin{bmatrix}K_1\ b\end{bmatrix}f^{(c)}$ with $a + b + c > d$ can be expressed as an ${\mathcal{A}}$-linear combination of elements with $a' + b' + c' \leq d$, proving that the proposed basis spans $S_{\mathcal{A}}(2,d)$.
• The basis $\{e^{(a)}\begin{bmatrix}K_1\ b\end{bmatrix}f^{(c)} \mid a + b + c \leq d\}$ is shown to be a free ${\mathcal{A}}$-basis of rank $d+1$ for the degree-zero part, and the full basis is constructed via the triangularity and spanning argument.