All Questions
14 questions with no upvoted or accepted answers
11
votes
0
answers
359
views
Representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$
Is there a nice reference for the finite dimensional (characteristic 0) representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$ and $\operatorname{PGL}_2(\mathbb Z/n\mathbb Z)$ for ...
- 7,746
8
votes
0
answers
366
views
Higher-dimensional generalization of Pink's theorem
Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...
- 26k
7
votes
0
answers
261
views
Invariant lattices of group representations over a $p$-adic field
Let $G$ be a finite group, $K$ be a $p$-adic field with an uniformizer $\pi$ and residue field $k \cong \Bbb F_q$, and $V$ be an irreducible representation of $G$ over $K$.
Let $X_{V}^G$ be the set ...
- 6,622
5
votes
0
answers
97
views
Is there a composite-order generalization of the homomorphism on Rep(Z/p) giving total dimension of Tate cohomology?
Let $p$ be a prime, let $\mathbb{Z}_p$ be the ring of $p$-adic integers, and let $G$ be a cyclic group of order $p$. It is rather well-known that finite rank $\mathbb{Z}_p$-free representations of $G$...
- 45.7k
5
votes
0
answers
219
views
Character tables of the p-core of the binary modular congruence group of p-power level
Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See The characters of binary modular congruence group, Bulletin of the
American Mathematical Society. 79 (1973), no. 4.), ...
- 2,396
4
votes
0
answers
176
views
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. https://math.stackexchange.com/questions/1428350/smooth-admissible-...
- 203
3
votes
0
answers
171
views
Conjugacy in metaplectic groups
Let $F$ be a non-Archimedean local field (characteristic 0) and $G=GL(2,F)$. Let $\tilde{G}$ be "the" metaplectic double cover of $G$ (defined using an explicit cocycle as in Gelbart's book (Weil's ...
- 185
3
votes
0
answers
130
views
Natural permutation representations of the orthogonal group of a lattice
Let $A\in M_n(\Bbb Z)$ be a positive definite matrix so it gives a metric on $\Bbb Z^n$. Let $G=O(q,\Bbb Z^n)$ be the subgroup of $GL_n(\Bbb Z)$ preserving the metric. For every positive integer $d$, $...
- 6,622
3
votes
0
answers
102
views
Localized at $p$ integral representations of finite elementary $p$-groups
Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...
- 14.1k
2
votes
0
answers
228
views
Satake correspondence for groups over finite field
I asked the same question in MSE, but I didn't get any answer. So I decided to post it here, too.
In Langlands' program, Satake correspondence gives a correspondence between unramified ...
- 2,215
2
votes
0
answers
63
views
Determining subgroup of finite group of Lie type from characteristic polynomials
Suppose you have $G$ a finite group of Lie type (say $\mathrm{Sp}_4( \mathbb{F}_5)$ as a case I particularly care about, but there are others.) Your friend picks a subgroup $H$ and selects random ...
- 2,429
1
vote
0
answers
174
views
What are the irreps in this canonical action of $\operatorname{PGL}_2(F_q)$?
Consider the permutation action of $\operatorname{PGL}_2(\mathbb F_q)$ on $\mathbb P^1(\mathbb F_q)$ by fractional linear transformations. We can consider the associated (complex) representation of ...
- 7,746
1
vote
0
answers
77
views
Existence of a certain direct summand
Let $P$ be a cyclic group of order $p^n$. Let $M$ be a $\mathbb{Z}_p[P]$-module and $M_p :=\mathbb{Q}_p\otimes M$ be the associated $\mathbb{Q}_p[P]$-module. When will $M_p$ have a direct summand ...
- 1,283
0
votes
0
answers
65
views
Higher-order obstructions in thin group orbits
Let $G$ be a finitely generated group acting on the integers $\mathbb{Z}$. Let $O_a = \{g \cdot a : g \in G\}$ be the orbit of an integer $a$ under this action. Assume that $O_a$ is a thin orbit, ...
