[Paper Review] On the good filtration dimension of Weyl modules for a linear algebraic group

Alison E. Parker|ArXiv.org|Jul 27, 2005

Advanced Algebra and Geometry11 references3 citations

TL;DR

This paper computes the Weyl filtration dimension of Weyl and induced modules for linear algebraic groups in positive characteristic, establishing that for regular weights, the Ext groups $<math>\operatorname{Ext}^i(\nabla(\lambda),\Delta(\mu))\cong k$ when $i = \operatorname{wfd}(\nabla(\lambda)) + \operatorname{wfd}(\nabla(\mu))$. This leads to exact formulas for projective and injective dimensions in generalized Schur algebras and global dimensions of $S(n,r)$ when $p>n$ or $S(mp,p)$, with the global dimension being twice the Weyl filtration dimension.

ABSTRACT

Let G be a linear algebraic group over an algebraically closed field of characteristic p whose corresponding root system is irreducible. In this paper we calculate the Weyl filtration dimension of the induced G-modules, abla(λ) and the simple G-modules L(λ), for λa regular weight. We use this to calculate some Ext groups of the form Ext^*( abla(λ),Δ(μ)), Ext^*(L(λ),L(μ)), and Ext^*( abla(λ), abla(μ)), where λ, μare regular and Δ(μ) is the Weyl module of highest weight μ. We then deduce the projective dimensions and injective dimensions for L(λ), abla(λ) and Δ(λ) for λa regular weight in associated generalised Schur algebras. We also deduce the global dimension of the Schur algebras for GL_n, S(n,r), when p>n and for S(mp,p) with m an integer.

Motivation & Objective

• To compute the Weyl filtration dimension (wfd) of induced modules $\nabla(\lambda)$ and simple modules $L(\lambda)$ for regular weights $\lambda$ in a linear algebraic group $G$ over an algebraically closed field of characteristic $p$.
• To determine the Ext groups $\operatorname{Ext}^*(\nabla(\lambda),\Delta(\mu))$, $\operatorname{Ext}^*(L(\lambda),L(\mu))$, and $\operatorname{Ext}^*(\nabla(\lambda),\nabla(\mu))$ for regular weights $\lambda, \mu$.
• To deduce the projective and injective dimensions of $L(\lambda)$, $\nabla(\lambda)$, and $\Delta(\lambda)$ in generalized Schur algebras for regular weights.
• To compute the global dimension of the Schur algebras $S(n,r)$ when $p>n$ and $S(mp,p)$ with $m\in\mathbb{N}$, and extend results to the quantum setting.

Proposed method

• Use translation functors introduced by Jantzen to analyze the structure of induced modules $\nabla(\lambda)$ and Weyl modules $\Delta(\mu)$ for regular weights.
• Establish that $\operatorname{Ext}^i(\nabla(\lambda),\Delta(\mu))\cong k$ precisely when $i = \operatorname{wfd}(\nabla(\lambda)) + \operatorname{wfd}(\nabla(\mu))$ for regular $\lambda, \mu$.
• Leverage the duality between Weyl and induced modules and the properties of translation functors to compute filtration dimensions.
• Apply the Weyl filtration dimension results to determine projective and injective dimensions of simple, induced, and Weyl modules in generalized Schur algebras.
• Extend the results to the quantum setting by replacing $p$ with $l$, the order of a root of unity, and using quantum translation functors and $l$-alcoves.
• Use the $\uparrow$-ordering on dominant weights and the structure of quasi-hereditary algebras to compute global dimensions of $q$-Schur algebras.

Experimental results

Research questions

• RQ1 What is the Weyl filtration dimension of the induced module $\nabla(\lambda)$ for a regular weight $\lambda$ in a linear algebraic group over a field of characteristic $p$?
• RQ2 How do the Ext groups $\operatorname{Ext}^*(\nabla(\lambda),\Delta(\mu))$ behave for regular weights $\lambda, \mu$?
• RQ3 What are the projective and injective dimensions of $L(\lambda)$, $\nabla(\lambda)$, and $\Delta(\lambda)$ in generalized Schur algebras for regular $\lambda$?
• RQ4 What is the global dimension of the Schur algebra $S(n,r)$ when $p>n$ or $S(mp,p)$ with $m\in\mathbb{N}$?
• RQ5 Can the results on Weyl filtration dimension and global dimension be extended to the quantum group setting and $q$-Schur algebras?

Key findings

• The Weyl filtration dimension of $\nabla(\lambda)$ for a regular weight $\lambda$ is $\operatorname{wfd}(\nabla(\lambda)) = (n-1)\left\lfloor\frac{r}{p}\right\rfloor$ for $\mathrm{GL}_n$ with $p>n$.
• For regular weights $\lambda, \mu$, $\operatorname{Ext}^i(\nabla(\lambda),\Delta(\mu))\cong k$ if and only if $i = \operatorname{wfd}(\nabla(\lambda)) + \operatorname{wfd}(\nabla(\mu))$.
• The projective dimension of $L(\lambda)$ in the generalized Schur algebra is $2\cdot\operatorname{wfd}(L(\lambda))$, and similarly for injective dimension.
• The global dimension of $S(n,r)$ is $2(n-1)\left\lfloor\frac{r}{p}\right\rfloor$ when $p>n$, and of $S(mp,p)$ is $2(m)(l-1)$ when $l=p$.
• The global dimension of the $q$-Schur algebra $S_q(n,r)$ is $2(n-1)\left\lfloor\frac{r}{l}\right\rfloor$ when $q$ is a primitive $l$th root of unity and $l>n$, and $2(l-1)m$ for $S_q(l,ml)$.
• The results extend to the quantum setting: the Weyl filtration dimension and global dimension of $S_q(n,r)$ depend only on $l$, not on the characteristic of the field.