Questions tagged [modular-representation-theory]
For questions about modular representation theory, the study of representations over a field of positive characteristic.
95
questions
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)...
