Questions tagged [modular-representation-theory]

For questions about modular representation theory, the study of representations over a field of positive characteristic.

Filter by
Sorted by
Tagged with
7
votes
2answers
159 views

Is it possible for the reduction modulo $p$ of an non-commutative semisimple algebra to be commutative?

Suppose that $I, X_1, \ldots, X_{d-1}$ are $n \times n$ matrices with integer entries whose $\mathbb{Z}$-span is a subalgebra of $\mathrm{Mat}_n(\mathbb{Z})$. Suppose that, thought of as a subalgebra ...
2
votes
0answers
42 views

Upper bound on decomposition numbers for the symmetric group in a block of weight $w$

The $p$-blocks of the symmetric group $S_n$ are labelled by pairs $(\gamma, w)$ where $\gamma$ is a $p$-core partition, $w \in \mathbb{N}_0$ is the weight of the block, and $|\gamma| + p w = n$. ...
-2
votes
0answers
195 views

$\bmod 2^k$ tricks for permanent vs determinant?

Given $M\in\{0,1\}^{n\times n}$ by the property of $-1=+1$ in $\mathbb F_2$ we infer $\mathsf{Permanent}(M)\bmod2\equiv\mathsf{Determinant}(M)\bmod2$ and hence we can compute $\mathsf{Permanent}(M)\...
3
votes
0answers
53 views

compactly induction of smooth modules over Hecke algebras

Let $G$ be a locally profinite group and $H$ its closed subgroup. It is well-known that a smooth representation of $G$ is identified with a smooth module over $\mathcal{H}(G)$, where $\mathcal{H}(G)$ ...
1
vote
1answer
293 views

Irreducible Representation of A_5

Knowing the fact that standard representation arising out of permutation representation of $A_5$ over $\mathbb{C}$ is irreducible and of degree $4$. What can we conclude about the irreducibility over ...
5
votes
1answer
94 views

Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$

Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
21
votes
1answer
716 views

How to see that the determinant of this matrix is nonzero for all primes?

I'm trying to show that $\sum_{i = 0}^{p-2} (i+1)^{-1} t^{i+n}$ where $0 \leq n \leq p-2$ spans the vector space $\mathbb{F}_p[t]/(1-t)^{p-1}$ as a rank $p-1$ module over $\mathbb{F}_p$. In other ...
1
vote
0answers
39 views

When is a block of a group algebra a trivial extension algebra?

Let $B$ be a block of a group algebra. See for example https://math.stackexchange.com/questions/229412/trivial-extension-of-an-algebra for the definition of a trivial extension algebra (which is ...
5
votes
0answers
120 views

Quiver and relations of $F\mathrm{SL}(2,q)$

$\DeclareMathOperator\SL{SL}$Let $q=p^n$ be a prime power and $F$ a field of characteristic two. Let $G=SL(2,q)$ the group of $2 \times 2$ special linear matrices over the field with $q$ elements with ...
4
votes
0answers
84 views

Which algebras of quaternion type do appear as blocks of group algebras?

A finite dimensional algebra $A$ is said to be of quaternion type (first defined by Erdmann?) when it is tame, symmetric and connected and furthermore the Cartan matrix is nonsingular and the stable ...
0
votes
0answers
54 views

Decomposition an $A$-module to irreducible ones

Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a complex algebra. Suppose that $V$ is an $A$-module. Under what proper condition(s) there are irreducible ...
5
votes
1answer
175 views

Representations of $\operatorname{Sp}(2g,\mathbb{Z}_3)$

Let $V$ be a $2g$-dimensional vector space over $\mathbb{Z}_3 := \mathbb{Z}/3\mathbb{Z}$. First, $\operatorname{Sp}(2g,\mathbb{Z}_3)$ acts on $\Lambda^2(V)$, and this decomposition is reducible, as ...
2
votes
0answers
101 views

Loewy structure of $S_4$

How to deduce the Loewy Structure of $kS_4$ where $k$ has characteristic 2. I can compute the Cartan matrix and Decomposition matrix with Brauer Characters without difficulties. But when it comes to ...
4
votes
0answers
79 views

No irreducible subrepresentations

Let $p$ be a prime number. In the paper The image of Colmez's Montreal functor, on the page 12, the author points out an interesting property: The smooth mod $p$ representation $\mathrm{c-Ind}^{GL_2(\...
4
votes
1answer
70 views

Field of definition for isomorphism classes of modular representations

Let $G$ be a finite group, and let $\sigma: G \to GL_n(k)$ be a (not necessarily irreducible) representation defined over an algebraically closed field $k$ of characteristic $p$. Let $\sigma^{(m)}$ ...
6
votes
1answer
126 views

Endomorphism ring of trivial source modules for abelian p-groups

Bernhard Böhmler  (who is also on MO) and myself had the following idea: Let $G$ be a finite group and $k$ a field of characteristic $p$ (algebraically closed when it is needed) such that $p$ divides ...
12
votes
0answers
312 views

Non-isomorphic groups with same character tables and different Brauer character tables

Let $A$ be a discrete valuation ring with perfect residue field $k$ of characteristic $p$ and field of fractions $K$ of characteristic $0$. Let $G$ and $H$ be two finite groups and assume that $K$ is ...
4
votes
1answer
202 views

Does any group action of $\mathbb{F}_p^\mathbb{N}$ on $\mathbb{F}_p^\mathbb{N}$ have non-trivial fixed points?

It is well-known that if $G$ is a finite $p$-group acting on a non-zero $\mathbb{F}_p$-vector space $V$, then $V^G \neq \{0\}$. My question is about a generalization of this result when $G = V = \...
3
votes
0answers
83 views

Reference for representations of $\text{GL}_n(\mathbb{Z}/p^k\mathbb{Z})$ over $\mathbb{F}_p$

I was looking for a reference on representations of $GL_n(\mathbb{Z}/p^k\mathbb{Z})$ over $\mathbb{F}_p$. In particular what the irreducible and indecomposable representations look like.
3
votes
1answer
132 views

Fourier transform on finite groups in characteristic $p>0$

Is there a Fourier theory for finite groups in characteristic $p>0$? Assume that $p$ divides the order $|G|$ of finite groups (or just work with $p$-groups), i.e., in a modular representation-...
0
votes
1answer
109 views

Faithful representation of group of order $p^4$

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book, "Theory of groups of finite order". The group ($\mathbb{Z}_{p^{2}}\rtimes \mathbb{Z}_{p^{}}) ...
3
votes
1answer
66 views

Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks

A version of Brauer's second main theorem is as follows: Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$. If $d_{\chi\mu}^x\neq 0$ and $\mu$ ...
11
votes
2answers
417 views

Relations between quantum groups at roots of unity, modular representation theory, and physics

I understand that quantum groups at roots of unity are related to physics because they are used in the construction of Reshetikhin-Turaev invariants, conjectured by Witten. Are there other relations ...
2
votes
0answers
75 views

Real non-principal 2-blocks for finite groups of Lie type

Is it true that each finite group of Lie type has a non-principal real $2$-block? Here the principal block is the one containing the trivial character and a block is real if it contains the complex ...
3
votes
0answers
61 views

Isomorphism of certain irreducible representations over finite fields

We are given a faithful representation of a cyclic group of order 5 $\rho: C_5=G \rightarrow End_{\mathbb{F}_3}(V) $ with $dim_{\mathbb{F}_3}V=8$ as vector space. It is also known that $V=U\oplus W$ ...
2
votes
1answer
223 views

When is there a subring of the complex numbers surjecting onto a given field of prime characteristic?

To make use of the Lie algebra action of $\mathsf{gl}_2(\mathbb{C})$ to establish a isomorphism in modular representation theory, I would like an answer to this question: Let $K$ be a field of ...
2
votes
1answer
123 views

Are there always 1-dimensional projective representations

Let $G$ be a finite $p$-group and $(\mathbb{F},\mathcal{O},\mathbb{K})$ a sufficiently large $p$-modular system. Furthermore let $\mu_{p^\infty}\leq \mathcal{O}^\times$ the group of roots of unity of $...
2
votes
1answer
90 views

2-quotient of integer partition

This question is mostly about understanding the notation used in the following article: Alex Eskin, Andrei Okounkov, Pillowcases and quasimodular forms, in: Victor Ginzburg (ed.), Algebraic Geometry ...
3
votes
1answer
136 views

Vanishing of first co-homology with coefficients modular representations of small dimension

Is the following true: For any $n$ there exists $p_0$ s.t. for any finite group $G$ of Lie type of rank $\leq n$ and characteristic $p\geq p_0$ and any (irreducible) $\mathbb F_p$ representation $V$ ...
11
votes
1answer
425 views

conductor formula

Let $\pi_p$ be an irreducible representation of $GL_2(\mathbb{Q}_p)$. Assume $\pi_p$ is ramified,hence it will have a positive conductor. Consider $sym^3(\pi_p)$ which is a representation of $GL_4(\...
5
votes
0answers
91 views

Extreme no loop conjecture for group algebras

Let $A=KG$ be a group algebra for a finite group $G$. Let $S$ be a simple $A$-module. The extreme no loop conjecture predicts that $Ext_A^1(S,S) \neq 0$ implies $Ext_A^i(S,S) \neq 0$ for infinitely ...
1
vote
1answer
64 views

Selfextensions for modules of commutative Frobenius algebras

Let $A$ be a commutative finite dimensional Frobenius algebra and $M$ a non-projective $A$-module. Can we have $Ext_A^i(M,M)=0$ for some $i>0$? Can we have $Ext_A^i(M,M)=0$ for some $i>0$ in ...
11
votes
0answers
475 views

Does $\mathrm{Ext}^1(M,M) \neq 0$ imply $\mathrm{Ext}^2(M,M) \neq 0$?

$\DeclareMathOperator{\Ext}{\operatorname{Ext}}$The first question is about group algebras: Question 1: Let $A=kG$ be a group algebra (with $G$ finite) and let $M$ be an indecomposable $A$-module. ...
10
votes
1answer
937 views

Gelfand's trick (Gelfand's lemma) in positive characteristic?

I came across this preprint that claims in Lemma 1.1 that Gelfand's trick (also known as Gelfand's lemma) only works in characteristic zero: Let $H < G$ be finite groups. Suppose we have an anti-...
2
votes
1answer
126 views

Representation of a finite group over a finite field from rational representations

Suppose $G$ is a the cyclic group of $n$ elements and $p$ is a prime not in $n$. $G$ has an action on the cyclotomic field as a $\mathbb{Q}$-vector space $\mathbb{Q}[X] / \langle \Phi_n(X) \rangle$ by ...
11
votes
0answers
242 views

Examples and non-examples of Tannakian $\infty$-categories

My definition of a Tannakian $\infty$-category is taken from this paper ("Tannaka duality over ring spectra" by James Wallbridge). (p. $53$, definition $7.9$) Let $R$ be an $E_{\infty}$-...
1
vote
0answers
122 views

Which set of compact objects generates the subcategory of a compactly generated stable model category?

I couldn't find any info on what set of compact objects generates the following subcategory: Let $k$ be a field of positive characteristic and let $G$ be either a finite group or a finite group ...
6
votes
1answer
365 views

first group cohomology for the standard representation of $S_n$ over $\mathbb{F}_2$

Let $g \geq 2$ be an integer and consider the symmetric group $S_n$ where $n = 2g+1$ or $n = 2g+2$ as a subgroup of the symplectic group $\mathrm{Sp}_{2g}(\mathbb{F}_2)$ via the standard ...
7
votes
1answer
239 views

Why is Nagao's theorem the “Module theoretic version of Brauer's second main theorem”?

Let $G$ be a finite group, $p\in\mathbb{P}$ a prime, $\mathbb{F}$ an algebraically closed field of characteristic $p$, and $D\leq G$ a $p$-subgroup. Brauer second main theorem states If $\chi\in ...
1
vote
0answers
52 views

To find $\dim(D^{\mu})=\dim\frac{S^{\mu}}{S^{\mu}\cap S^{\mu\perp}}$

Let $S_6$ be the symmetric group of degree $6$ and $F$ be any finite field of characteristic $2.$ Then $2$-regular partition of $6$ are $(5,1)$, $(4,2)$ and $(3,2,1)$ . I have to find $$\dim(D^{\mu})=\...
1
vote
0answers
28 views

Decomposition numbers of characteristic 0 Ariki-Koike algebras for q=0 or q generic

Consider a complex Ariki-Koike algebra $H:=H(q;Q_1,\dots, Q_\ell)$ for some invertible elements $q, Q_1, \dots, Q_\ell$ of $\mathbb{C}$, i.e. an associative $\mathbb{C}$-algebra with generators $T_0, \...
5
votes
1answer
186 views

Questions on group and Nakayama algebras from a book

Recall that a Nakayama algebra (also called serial algebras in the literature) is an algebra such that every indecomposable module has a unique composition series. In the book "Classical artinian ...
9
votes
0answers
172 views

Modular version of Mednykh's formula?

Let $G$ be a finite group and $\Sigma_g$ a closed Riemann surface of genus $g$. Then Mednykh's formula states $\frac{\left|\mathrm{Hom}(\pi_1(\Sigma_g),G)\right|}{\left|G\right|} = \frac{1}{\left|G\...
2
votes
0answers
46 views

Branching rule for degenerate cyclotomic Hecke algebras

Grojnowski and Vazirani showed that there is a crystal isomorphism between the crystal defined by modular branching of simple modules of Ariki-Koike algebras and that of an integral highest weight ...
1
vote
0answers
39 views

Why are the definitions of i-good nodes of a multipartition equivalent?

Let $e\geq 2$ and $0\leq i\leq e-1$. For a multipartition $\lambda\vdash_\ell n$ of $n $ one can define the notion of an $i$-good box of the Young diagram of $\lambda$. But there are seemingly ...
3
votes
0answers
141 views

Questions about ``$p$-canonical basis" for $\widehat{\mathfrak{sl}_n}$ module (wedge power of natural representation)

Let $p$ be a prime number. Consider the natural representation of the affine Lie algebra $\widehat{\mathfrak{sl}_p}$, defined as follows. $$A = \bigoplus_{i=1}^N \mathbb{C}a_i; \qquad \text{nat}_p = A ...
1
vote
0answers
97 views

$G$-invariant bilinear maps

Let $G$ be a finite group and $M$ a finitely generated $\mathbb{Z}G$-module. Then $M \otimes k$ is a representation of $G$ for any field $k$. I am interested in the number of ways we can turn $M \...
11
votes
1answer
331 views

Can we glue characteristic 0 and characteristic p representations of a finite group given equality of (Brauer) characters?

Suppose I have a prime $p$ and a finite group $G$ together with representations $\sigma: G \to GL_n(\mathbb{Q}_p)$ and $\pi: G \to GL_n(\mathbb{F}_p)$. My question is: When does there exist a ...
4
votes
1answer
88 views

Sufficient conditions for secondary invariants

Let $G$ be a finite group, $k$ be a field whose characteristic divides $|G|$, and $\rho:G\hookrightarrow\operatorname{GL}_n(k)$ be a faithful representation of $G$. Let $V$ be a $k$-space of dimension ...
4
votes
0answers
119 views

subgroups of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose mod-2 image is the symmetric group

Let $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ be a closed subgroup of the symplectic group over the $2$-adic integers whose image under the mod-$2$ homomorphism $\pi : \mathrm{Sp}_{2g}(\mathbb{Z}_2)...