Search Results

You would have an answer to your question if you could classify pairs of elements in $GL_n(\mathbf F_q)$ up to simultaneous conjugacy. This is the notorious matrix pair problem, which is the quintesse …

For $l=2$, the only cases where the extension splits are $n=2$ and $p=2,3$ and $n=3$ and $p=2$. In her PhD thesis, on page 3, Pooja Singla attributes this to [Chih Han Sah. Cohomology of split group e …

Here is an outline for odd order abelian $p$-groups: The main point is that every non-zero element in a field with at least three elements is a sum of two non-zero elements. Using this, you can show …

Victor's answer shows that it is important to understand the action of $Out(G)$ on the conjugacy classes of $G$. This can be interesting even in the abelian case, where the problem amounts to calculat …

For a field, this is given by the rational canonical form (see Section 7.2 of Hoffman and Kunze's Linear Algebra, for example). Even in this case, the trace and characteristic polynomial are quite wea …

The problem of enumerating subgroups of a finite abelian group is both non-trivial and interesting. It is also worthwhile to study this set as a lattice. As far as I know, the reference "Subgroup latt …

According to Djokovic and Maizan, the Specht module $V_{(3, 1, 1)}$ of $S_5$ is monomial. This is a representation of dimension $6$, induced from a representation of dimension $3$ of $A_5$. Since $A_5 …

The proof of Mackey's theorem on intertwiners actually tells you how to construct the endomorphism algebra of an induced representation, not just its dimension. So, if you work a little harder, you ma …

To a finite $p$-group, we can associate two vectors $(v_0,v_1,\dotsc)$: The class vector - $v_i$ is the number of conjugacy classes of order $p^i$. The character vector - $v_i$ is the number of comp …

I found the paper of Chriss and Khuri-Makdisi (Chriss, Neil; Khuri-Makdisi, Kamal. On the Iwahori-Hecke algebra of a $p$-adic group. Internat. Math. Res. Notices 1998, no. 2, 85--100.) quite helpful. …

This is answered in Theorem 4 of my paper Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups, with Geetha Thangavelu, which is published in Algebras and Representation Theory, an …

Bruhat decomposition over $\mathbf Z/p^r\mathbf Z$ is precisely the problem we looked at in this paper. We defined several invariants of double cosets, and classified the pairs $(n,k)$ for which, when …

I believe that your conjecture is equivalent to Theorem 3.5 in the paper Locally Compact Abelian Groups with Symplectic Self-duality, Advances in Mathematics, volume 225, pages 2429-2454, 2010.

Let's say you want to compute the Hall polynomial $g^\lambda_{(r),\mu}(p)$. According to [Dutta and Prasad, Degenerations and orbits in finite abelian groups], the orbits under the automorphism group …

The answer is a special case of Young's rule. In my book, I give a very simple method for the slightly easier case where $r=0$. In that case we have: $$ \mathrm{Ind}_{S_k\times S_l}^{S_n} = \bigoplus_ …