Given an n×n matrix c over a unitary ring R, the centralizer of in full Mn(R) is called principal ring, denoted by Sn(c,R). We investigate its structure and prove: (1) If invertible with c-free point, or if R has no zero-divisors Jordan-similar all eigenvalues center then separable Frobenius extension Sn(c,R) sense Kasch. (2) integral domain matrix, cellular R-algebra Graham Lehrer. In particul...

Suppose G(R) is a real reductive group, with L-group ∨ G, and φ : WR → ∨ G is an L-homomorphism. There is a close relationship between the Lpacket associated to φ and characters the component group of the centralizer of the image of φ. This was originally observed by Shelstad [10], and further refined by Arthur, Kottwitz, Langlands and Shelstad. This is reinterpreted in [3], to (among other thi...

The fusion orbit category F‾C(G) of a discrete group G over collection C is the whose objects are subgroups H in C, and morphisms H→K given by G-maps G/H→G/K modulo action centralizer CG(H). We show that higher limits can be computed using hypercohomology spectral sequences coming from Dwyer G-spaces for normalizer decompositions G. If realizing saturated system F, then these give two converge ...

for a finite group $g$‎, ‎let $cent(g)$ denote the set of centralizers of single elements of $g$‎. ‎in this note we prove that if $|g|leq frac{3}{2}|cent(g)|$ and $g$ is 2-nilpotent‎, ‎then $gcong s_3‎, ‎d_{10}$ or $s_3times s_3$‎. ‎this result gives a partial and positive answer to a conjecture raised by a‎. ‎r‎. ‎ashrafi [on finite groups with a given number of centralizers‎, ‎algebra‎ ‎collo...

My research focuses on topics related to unipotent classes and Springer theory for algebraic groups. These concepts are intimately related to the representation theory of Weyl groups. I have recently also become interested in complex reflection groups, which can be viewed as generalizations of Weyl groups. Combinatorial methods and algorithms have a long history as a means for studying all of t...

To 2-categorify the theory of group representations, we introduce the notions of the 3-representation of a group in a strict 3-category and the strict 2-categorical action of a group on a strict 2-category. We also 2-categorify the concept of the trace by introducing the 2-categorical trace of a 1-endomorphism in a strict 3-category. For a 3-representation ρ of a group G and an element f of G, ...

We introduce the notions of proto-complete, complete, complete⁎ and strong-complete objects in pointed categories. show under mild conditions on a exact protomodular category that every proto-complete (respectively complete) object is product an abelian object. This together with observation trivial group only complete recovers theorem Baer classifying groups. In addition we generalize several ...