Algebra & Combinatorics Seminar:   2023–24

Abstract. Let $\mathfrak g$ be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra $\mathfrak h$. We prove that unique factorization holds for tensor products of parabolic Verma modules. We prove more generally a unique factorization result for products of characters of parabolic Verma modules when restricted to certain subalgebras of $\mathfrak h$. These include fixed point subalgebras of $\mathfrak h$ under subgroups of diagram automorphisms of $\mathfrak g$. This is joint work with K.N.Raghavan, R.Venkatesh and S.Viswanath.

Abstract. The Alpha invariant of a complex Fano manifold was introduced by Tian to detect its K-stability, an algebraic condition that implies the existence of a Kähler–Einstein metric. Demailly later reinterpreted the Alpha invariant algebraically in terms of a singularity invariant called the log canonical threshold. In this talk, we will present an analog of the Alpha invariant for Fano varieties in positive characteristics, called the Frobenius-Alpha invariant. This analog is obtained by replacing "log canonical threshold" with "F-pure threshold", a singularity invariant defined using the Frobenius map. We will review the definition of these invariants and the relations between them. The main theorem proves some interesting properties of the Frobenius-Alpha invariant; namely, we will show that its value is always at most 1/2 and make connections to a version of local volume called the F-signature.

Abstract. Associated to every reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, obtained by killing all "sufficiently long" braid words, as well as some integer power of each generator. These include usual nil-Coxeter algebras, nil-Temperley–Lieb algebras, and their variants, and lead to symmetric semigroup module categories which necessarily cannot be monoidal.

Motivated by the classical work of Coxeter (1957) and the Broue–Malle–Rouquier freeness conjecture, and continuing beyond the previous work of Khare, we attempt to obtain a classification of the finite-dimensional nil-Hecke algebras for all reflection groups $W$. These include the usual nil-Coxeter algebras for $W$ of finite type, their "fully commutative" analogues for $W$ of FC-finite type, three exceptional algebras (of types $F_4$,$H_3$,$H_4$), and three exceptional series (of types $B_n$ and $A_n$, two of them novel). We further uncover combinatorial bases of algebras, both known (fully commutative elements) and novel ($\overline{12}$-avoiding signed permutations), and classify the Frobenius nil-Hecke algebras in the aforementioned cases. (Joint with Apoorva Khare.)