Search Results

$\newcommand{\GFq}{\mathbf F_q}$ Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial $p_n( …

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 …

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 …

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 …

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 …

The space that you seek is the two-sided ideal in $\mathbb C[G]$ generated by the character of $\pi$ (see details below). This follows from the explicit Wedderburn decomposition of $\mathbb C[G]$. If …

The motivation for studying Iwahori-Hecke algebras (the convolution type) is the study of the unramified principal series representations. The details can get lengthy, but it is all in a paper of Bore …

Jeff Adams has comprehensive notes on his website: http://www.math.umd.edu/~jda/characters/characters.pdf The irreducible characters of the groups SL(2), PGL(2), GL(2) and PSL(2) over finite fields a …

It is not difficult to show that the number of conjugacy classes in the alternating group $A_n$ is given by classes in the alternating group = no. of even partitions + no. of self-transpose partit …

A group is said to be a $Q$-group if the character of any complex representation is rational valued. A well-known internal characterization of $Q$-groups is the following: $G$ is a $Q$-group if an …

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_ …

Here is just one example (I know there are others too): Just as representations of the Hecke algebra associated to a Weyl group correspond to representations of a finite group of Lie type which are i …

If you like combinatorics, you may enjoy learning about the representations of $S_n$ by reading Chapter 7 of Stanley's Enumerative Combinatorics, Volume 2.

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. …