I will explain how Dirac operators provide precious information about geometric and algebraic aspects of representations of real Lie groups. In particular, we obtain an explicit realisation of representations, leading terms in the asymptotics of characters and a precise connection with nilpotent orbits.

# Past Algebra Seminar

We will discuss some recent results with Martin Bridson about

Sidki's construction X(G). In particular, if G is a finitely presented

group then X(G) is a finitely presented group. We will discuss as well the

result that if G has polynomial isoperimetric function and the maximal

metabelian quotient of G is virtually nilpotent then X(G) has polynomial

isoperimetric function. Part of the arguments we will use have homological

nature.

Abstract regular polytopes are finite quotients of Coxeter complexes

with string diagram, satisfying a natural intersection property, see

e.g. [MMS2002]. They arise in a number of geometric and group-theoretic

contexts. The first class of such objects, beyond the

well-understood examples coming from finite and affine Coxeter groups,

are locally toroidal cases, e.g. extensions of quotients of the affine

F_4 complex [3,3,4,3]. In 1996 P.McMullen & E.Schulte constructed a

number of examples of locally toroidal abstract regular polytopes of

type [3,3,4,3,3], and conjectured completeness of their list. We

construct counterexamples to the conjecture using a Y-shaped

presentation for a subgroup of the Monster, and discuss various

related questions.

For a reductive group $ G $, Steinberg established a map from the Weyl group to nilpotent $ G $-orbits using momentmaps on double flag varieties. In particular, in the case of the general linear group, he re-interpreted the Robinson-Schensted correspondence between the permutations and pairs of standard tableaux of the same shape in terms of product of complete flags.

We generalize his theory to the case of symmetric pairs $ (G, K) $, and obtained two different maps. In the case where $ (G, K) = (\GL_{2n}, \GL_n \times \GL_n) $, one of the maps is a generalized Steinberg map, which induces a generalization of the RS correspondence for degenerate permutations. The other is an exotic moment map, which maps degenerate permutations to signed Young diagrams, i.e., $ K $-orbits in the Cartan space $ (\lie{g}/\lie{k})^* $.

We explain geometric background of the theory and combinatorial procedures which produces the above mentioned maps.

This is an on-going joint work with Lucas Fresse.

Signed permutation modules of symmetric groups and Iwahori-hecke algebras

There are many interesting problems surrounding the unit group U(RG) of the ring RG, where R is a commutative ring and G is a finite group. Of particular interest are the finite subgroups of U(RG). In the seventies, Zassenhaus conjectured that any u in U(ZG) is conjugate, in the group U(QG), to an element of the form +/-g, where g is an element of the group G. This came to be known as the "(first) Zassenhaus conjecture". I will talk about the recent construction of a counterexample to this conjecture (this is joint work with L. Margolis), and recent work on related questions in the modular representation theory of finite groups.

Abstract: In 2016 Ayyer, Prasad and Spallone proved that the restriction to

S_{n-1} of any odd degree irreducible character of S_n has a unique irreducible

constituent of odd degree.

This result was later generalized by Isaacs, Navarro Olsson and Tiep.

In this talk I will survey some recent developments on this topic.

Decomposition (aka unital 2-Segal) spaces are simplicial ∞-groupoids with a certain exactness property: they take pushouts of active (end-point preserving) along inert (distance preserving) maps in the simplicial category Δ to pullbacks. They encode the information needed for an 'objective' generalisation of the notion of incidence (co)algebra of a poset, and motivating examples include the decomposition spaces for (derived) Hall algebras, the Connes-Kreimer algebra of trees and Schmitt's algebra of graphs. In this talk I will survey recent activity in this area, including some work in progress on a categorification of (Hopf) bialgebroids.

This is joint work with Imma Gálvez and Joachim Kock.

Let $\mathfrak g$ be a semisimple Lie algebra. A $\mathfrak g$-algebra is an associative algebra $R$ on which $\mathfrak g$ acts by derivations. There are several significant examples. Let $V$ a finite dimensional $\mathfrak g$ module and take $R=\mathrm{End} V$ or $R=D(V)$ being the ring of derivations on $V$ . Again take $R=U(\mathfrak g)$. In all these cases $ S=U(\mathfrak g)\otimes R$ is again a $\mathfrak g$-algebra. Finally let $T$ denote the subalgebra of invariants of $S$.

For the first choice of $R$ above the representation theory of $T$ can be rather explicitly described in terms of Kazhdan-Lusztig polynomials. In the second case the simple $T$ modules can be described in terms of the simple $D(V)$ modules. In the third case it is shown that all simple $T$ modules are finite dimensional, despite the fact that $T$ is not a PI ring, except for the case $\mathfrak g =\mathfrak {sl}(2)$.

A law for a group G is a non-trivial equation satisfied by all tuples of elements in G. We study the length of the shortest law holding in a finite group. We produce new short laws holding (a) in finite simple groups of Lie type and (b) simultaneously in all finite groups of small order. As an application of the latter we obtain a new lower bound on the residual finiteness growth of free groups. This talk is based on joint work with Andreas Thom.