I will present some interesting computations concerning polynomial and rational invariants of nilpotent Lie algebras. I will say more about standard filiform Lie algebras which appear to have the highest level of complication among the small-dimensional algebras. I will outline an implementable algorithm for the computation of generators of the field of rational invariants.
We first give some background on the Poincaré series of the algebra of generic matrices and its associated trace ring, and then focus on some recent work, including a conjecture for the denominator of the one variable series for the trace rings. Time permitting we will also say a bit about traces of direct sums.
SLn webs first emerged in invariant theory and have a recent reformulation by Cautis-Kamnitzer-Morrison (2014). A collection of these webs form a basis of the Specht modules for the symmetric groups. On the other hand, classical construction of the Specht modules uses the polytabloids basis parameterized by standard Young tableaux. Russell-Tymoczko (2020) showed that the transitioning matrix from the polytabloid basis to the web basis is unitriangular. We further proved their conjecture that the upper-triangular entries are positive. The talk will be mostly focused on SL2 webs, with some preliminary results on SL3 webs.
Motivated by a recent Diophantine transport problem about how to transport profitably a group of persons or objects, we survey classical facts about solving systems of linear Diophantine equations and inequalities in non-negative integers. We emphasize on the method of Elliott from 1903 and its further development by MacMahon in his 'Ω-Calculus' or Partition Analysis. Then we show how this approach can be used to solve problems in classical and non-commutative invariant theory and theory of algebras with polynomial identities.
An LMS online lecture course in Vinberg theory.
In recent years, Vinberg theory of graded Lie algebras has become relevant in many areas of number theory, from arithmetic statistics (e.g., in the work of Romano-Thorne) to the local Langlands correspondence (e.g., in the work of Reeder-Yu). These lectures will provide the algebraic background for number theory students to engage with research involving graded Lie algebras. We'll start by discussing some of the relevant aspects of the invariant theory of Lie algebras, including the Chevalley restriction theorem and the pioneering work of Kostant on invariant rings. We'll then define graded Lie algebras and look at the graded analogues of these theorems, based on work of Vinberg. Time permitting, we'll look at Slodowy slices and applications to families of algebraic curves. These lectures should give number theory students sufficient background to read, for example, Thorne's paper Vinberg's representations and arithmetic invariant theory and other related papers. But the lectures will also be a useful introduction to some beautiful aspects of Lie theory for students in algebra and representation theory. I'll assume students have some knowledge of Lie algebras, but I will review relevant background and provide examples throughout the lectures.
Let G be a finite subgroup of GLn(K) for a field K whose characteristic does not divide the order of G. The group G acts linearly on the polynomial ring S in n variables over K. When G is generated by reflections, then the discriminant D of the group action of G on S is a hypersurface with a singular locus of codimension 1. In this talk we give a natural construction of a noncommutative resolution of singularities of the coordinate ring of D as a quotient of the skew group ring A = S ∗ G by the idempotent e corresponding to the trivial representation. We will explain how this can be seen in some sense as a McKay correspondence for reflection groups.
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