arXiv subjects

William D. Hardesty

Publications and source records attributed to William D. Hardesty.

4 recordsLinked to original sources

arXiv2019

Calculations with graded perverse-coherent sheaves

In this paper, we carry out several computations involving graded (or $\mathbb{G}_{\mathrm{m}}$-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper, we compute the weight of the $\mathbb{G}_{\mathrm{m}}$-action on certain normalized (or "canonical") simple objects, confirming an old prediction of Ostrik. In the second part of the paper, we explicitly describe all simple perverse coherent sheaves for $G = PGL_3$, in every characteristic other than 2 or 3. Applications include an explicit description of the cohomology of tilting modules for the corresponding quantum group, as well as a proof that $\mathsf{PCoh}^{\mathbb{G}_{\mathrm{m}}}(\mathcal{N})$ never admits a positive grading when the characteristic of the field is greater than 3.

math.RT↗

arXiv2018

On the Existence of Mock Injective Modules for Algebraic Groups

Let $G$ be an affine algebraic group scheme over an algebraically closed field $k$ of characteristic $p>0$, and let $G_r$ denote the $r$-th Frobenius kernel of $G$. Motivated by recent work of Friedlander, the authors investigate the class of mock injective $G$-modules, which are defined to be those rational $G$-modules that are injective on restriction to $G_r$ for all $r\geq 1$. In this paper the authors provide necessary and sufficient conditions for the existence of non-injective mock injective $G$-modules, thereby answering a question raised by Friedlander. Furthermore, the authors investigate the existence of non-injective mock injectives with simple socles. Interesting cases are discovered that show that this can occur for reductive groups, but will not occur for their Borel subgroups.

math.GR↗

arXiv2016

Support varieties of line bundle cohomology groups for SL3 (k)

Let $G= SL_3(k)$ where $k$ is a field of characteristic $p > 0$ and let $λ\in X(T)$ be any weight with corresponding line bundle $\mathscr{L}(λ)$ on $G/B$. In this paper we compute the support varieties for all modules of the form $H^i(λ):= H^i(G/B, \mathscr{L}(λ))$ over the first Frobenius kernel $G_1$. The calculation involves certain recursive character formulas given by Donkin which can be used to compute the characters of the line bundle cohomology groups. In the case where $λ$ is a $p$-regular weight and $M=H^i(λ)\neq 0$ for some $i$, these formulas are used to show that any $p^{th}$ root of unity $ζ$ is not a root of the generic dimension of $M$. To handle the case where $λ$ is not $p$-regular, we employ techniques similar to those used by Drupieski, Nakano and Parshall to show that the module $H^i(λ)$ is not projective over $G_1$ whenever it is nonzero and $λ$ lies outside of the Steinberg block.

math.RT↗

arXiv2015

On support varieties and the Humphreys conjecture in type $A$

Let $G$ be a reductive algebraic group scheme defined over $\mathbb{F}_p$ and let $G_1$ denote the Frobenius kernel of $G$. To each finite-dimensional $G$-module $M$, one can define the support variety $V_{G_1}(M)$, which can be regarded as a $G$-stable closed subvariety of the nilpotent cone. A $G$-module is called a tilting module if it has both good and Weyl filtrations. In 1997, it was conjectured by J.E. Humphreys that when $p\geq h$, the support varieties of the indecomposable tilting modules coincide with the nilpotent orbits given by the Lusztig bijection. In this paper, we shall verify this conjecture when $G=SL_n$ and $p > n+1$.

math.RT↗