Draisma recently proved that finite length polynomial representations of the infinite general linear group GL are topologically GL-noetherian, i.e., the descending chain condition holds for GL-stable closed subsets. The scheme-theoretic variant of this theorem is a major open problem in the area. I will briefly outline the rich history of this problem and provide a negative answer in characteristic 2.
Let G be a simple, simply connected algebraic group defined over a field of positive characteristic p, Gr be its rth Frobenius kernel, and, for q = pr, G(q) denotes the group of rational points over a field with q elements.
Motivated by work of Curtis and Steinberg, who showed that the simple Gr- and the simple G(q)-modules can be lifted to G, Humphreys and Verma conjectured that the projective covers of the simple Gr-modules also afford a G-module structure. Donkin later refined this conjecture by suggesting that these Gr-projectives lift uniquely to G in the form of tilting modules.
Ballard and Jantzen verified Donkin’s Tilting Module Conjecture for primes that are roughly twice the Coxeter number of the underlying root system or larger. But it was shown by Nakano and his collaborators that the conjecture fails in general. Counterexamples exist for all root systems with the exception of types B2, where the conjecture holds, and type A, where the conjecture remains completely open.
In this talk we delve into the rich history of these and closely related conjectures and report on their current status.
The quest to find a character formula for the simple modules of a reductive algebraic group in positive characteristic took an unexpected turn roughly a decade ago when Williamson found a large number of counterexamples to the Lusztig Conjecture. Since then, the path to the simple characters has gone through the characters of the indecomposable tilting modules, thanks to the work of Riche and Williamson. However, the combinatorics required for determining all tilting characters are quite complicated, and the vast majority of these characters are not necessary to determine the simple characters. This talk is based on our pursuit of a more simplistic model in terms of what we’ve called the 'Steinberg quotient' of special tilting characters.
We discuss support variety theory for quasireductive algebraic supergroups, i.e. supergroups with reductive even part over complex numbers. The corresponding categories of representations are Frobenius and share many properties of representations of finite groups in positive characteristic. It is desirable to describe Balmer spectrum of related triangulated symmetric monoidal categories. Our approach involves so called homological odd elements and certain tensor functors associated to them. On the way we encounter analogues of p-groups and Sylow subgroups for supergroups. We prove projectivity detection for our support theory and present other related results. We also explore connections with homological support theory developed by B. Boe, J. Kujawa and D. Nakano.
We investigate infinite-dimensional modules for a linear algebraic group 𝔾 over a field of positive characteristic p. For any subcoalgebra C ⊂ 𝒪(𝔾) of the coordinate algebra of 𝔾, we consider the abelian subcategory CoMod(C) ⊂ Mod(𝔾) and the left exact functor (−)C : Mod(𝔾) → CoMod(C) that is right adjoint to the inclusion functor. The class of cofinite 𝔾-modules is introduced using finite-dimensional subcoalgebras of 𝒪(𝔾). We categorify a construction of Hardesty-Nakano-Sobaje, thereby supplementing cofinite type in providing invariants for proper mock injective 𝔾-modules.
In type A, the Macdonald polynomials and the integral from Macdonald polynomials are related by a plethystic transformation. We interpret this plethystic transformation geometrically as a relationship between nilpotent parabolic Springer fibres and nilpotent Lusztig varieties. This points the way to a generalization of modified Macdonald polynomials and integral form Macdonald polynomials to all Lie types. But these generalizations are not polynomials, they are elements of the Iwahori-Hecke algebra of the finite Weyl group. This work concerns the generalization of, and connection between, a 1997 paper of Halverson-Ram (which counts points of nilpotent Lusztig varieties over a finite field) and a 2017 paper of Mellit (which counts points of nilpotent parabolic affine Springer fibres over a finite field).
In this talk, we first present our study on the number of partitions of a positive integer m into at most n parts in a given set A. We prove that such a number is bounded by the nth Fibonacci number F(n) for any m and some family of sets A including sets of powers of an integer. Then we use this result to estimate the cohomology space of the simple algebraic group SL2 with coefficients in Weyl modules. In particular, let k be an algebraically closed field of prime characteristic p and V(m) the Weyl SL2-module of highest weight m. We show that for p ≥ 5, dim Hn(SL2,V(m))≤ F(n+1) for all m,n ≥ 0.
We discuss linkage principles and blocks for general linear, ortho-symplectic, and periplectic supergroups over fields of positive characteristics. In the end, we describe the strong linkage principle and blocks for the queer supergroup Q(2).
In the study of cohomology of finite group schemes it is well known that nilpotence theorems play a key role in determining the spectrum of the cohomology ring.
Balmer recently showed that there is a more general notion of a nilpotence theorem for tensor triangulated categories through the use of homological residue fields and the connection with the homological spectrum. The homological spectrum can be viewed as a topological space that realizes the Balmer spectrum in a concrete way.
Let 𝔤=𝔤0 ⊕ 𝔤1 be a Type I classical Lie superalgebra with an ample detecting subalgebra. In this talk, the speaker will consider the tensor triangular geometry for the stable category of finite-dimensional Lie superalgebra representations: stab(ℱ(𝔤,𝔤0)).
The localizing subcategories for the detecting subalgebra 𝔣 are classified which answers a question of Boe, Kujawa and Nakano. As a consequence of these results, the we prove a nilpotence theorem and determine the homological spectrum for the stable module category of ℱ(𝔣,𝔣0).
The orbit structure of the reductive group G0 on 𝔤1 where Lie G0=𝔤0 along with the results for the detecting subalgebra is used to prove a nilpotence theorem for stab(ℱ(𝔤,𝔤0)) and to determine the homological spectrum in this case.
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