Smooth admissible representations, Hom, tensor and extension of scalars

(Remark: This has previously been posted on math.stackexchange, but I believe it might be suitable for this site as well. ) Let G be a locally profinite group, and consider V and W smooth admissible representations of G over some field F (or char. 0). Let E/F be any field extension. I’d like to … Read more

Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let G be a finite group,let Rep(G) denote the category of finite dimensional representations over C. Let V,W be representations of G in Rep(G). One can define a bilinear form on Rep(G) or inner product in K0(Rep(G)) (in Teleman’s notes) as dimCHom(V,W)G which is G invariant of Hom(V,W).Then there … Read more

Exotic 2-adic lifts of mod $2$ Steinberg idempotent

Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices. The (conjugate) Steinberg idempotent is defined to be $$e_n’=\frac{1}{q_n}\Sigma _{g\in \Sigma _n}sgn(g)g \Sigma _{g\in B _n}g \in Z_{(2)}[Gl_n(Z/2)]$$ where $q_n$ is an appropriate odd number, $sgn$ is the signature of the permutation. Now, let’s … Read more

Properties of Higman’s group

The infinite group of Higman which has no finite quotient is given by the presentation (with 4 generators and 4 relations): G=⟨ai,i∈Z/4Z∣aiai+1(mod 4)a−1i=a2i+1⟩. Its main feature (proved by Higman) is that [it is infinite and] it has no finite quotients. Another important feature (proved by Schupp, see this post) is that it is SQ-universal (:=”every countable … Read more

(Double) Crystal reflection operators on SSYTs

I am not that familiar with the language of crystals, but this is what I know: Let $SSYT(\lambda, \mu)$ be the set of semi-standard Young tableaux with shape $\lambda$ and weight $\mu$. There are crystal operators, $e_i$, $f_i$ that preserves the shape, but converts one box of content $i$ to a box with content $i+1$ … Read more

Seeking an unpublished manuscript by Tetsuro Okuyama

Several papers in representation theory attribute the notion of relatively projective modules to Tetsuro Okuyama’s manuscript “A generalization of projective covers of modules over finite group algebras.” I haven’t been able to find this by googling. Can anyone point me to a copy (or send me one)? Thanks! Answer AttributionSource : Link , Question Author … Read more

Fibers of torus equivariant moment maps

Given a closed (possibly singular) projective variety V with a symplectic structure and a torus action, there is a moment map μ:V→Lie(T)∗. Note that the dimension of T could be much smaller than the dimension of V. How much can we say about the fibers of this moment map μ? Any references? I am most … Read more

Shifts in the decomposition of Bott-Samelson bimodules

Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that $W\subset\operatorname{GL}(V)$ is a finite reflection group, and let $S\subset W$ be a set of simple reflections so that $(W,S)$ is a finite … Read more

Compactly supported distributions as a projective G-module

For a Lie group G and a locally convex space V let E(G,V) be the locally convex space of smooth functions from G to V, and accordingly E′c(G,V) the space of compactly supported distributions. A G-module V is called differentiable if v→(g→gv) defines a continuous map from V to E(G,V) for all v∈V. In particular … Read more

Short proof of the classification of representation-finite symmetric algebras up to stable equivalence

Assume K is an algebraically closed field and A a finite dimensional K-algebra. Assume additionally that A is symmetric and representation-finite. Then one has the following classification of such algebras up to stable equivalence( source without proof section 3.14 in: Andrzej Skowroński – Selfinjective algebras: finite and tame type): A is stable equivalent to a … Read more