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

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6
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1answer
151 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
50 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
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0answers
53 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
191 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
109 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
72 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
131 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
402 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(\...
0
votes
0answers
63 views

p-groups embedded into Sylow subgroups

Let $p$ be a prime number, $q$ a power of $p$ and $P$ be a finite $p$-group. $P$ is isomorphic to a subroup of p-Sylow subgroup of the symmetric group $S_{\mid P\mid}$ (Theorem of Cayley) the ...
4
votes
0answers
80 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
62 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 $i>0$...
9
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0answers
323 views

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

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. Does $Ext_A^1(M,M) \neq 0$ imply $Ext_A^2(M,M) \...
10
votes
1answer
662 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
104 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 ...
10
votes
0answers
213 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 $E_{\infty}$-ring, $C$ a ...
1
vote
0answers
100 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
315 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
207 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
51 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
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0answers
26 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
179 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 ...
8
votes
0answers
152 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
45 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
135 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
96 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
321 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
85 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
117 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)...
10
votes
3answers
457 views

Low-dimensional irreducible 2-modular representations of the symmetric group

I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even ...
1
vote
0answers
45 views

Largest almost quasisimple group that acts on a spin module

I'd like to know the largest almost quasisimple group $G$ which acts faithfully and irreducibly on the spin module for an odd dimensional orthogonal group $\Omega(2k+1,q)$, or on each of the two half-...
5
votes
2answers
338 views

Module with indecomposable and decomposable reductions mod $p$

Let $G$ be a finite group and $p$ a prime dividing the order of $G$. Let $V$ be an irreducible $\mathbb{C}[G]$-module. Let $F$ be the finite field $\mathbb{Z} / p \mathbb{Z}$. Suppose that there ...
13
votes
0answers
209 views

Group homology $\mathrm{SL}_2$ acting on $\mathrm{Sym}^g$

Let $k$ be a field. We write $\mathrm{Sym}^g(k^2)$ for the $g$-th symmetric power of the (a?) standard representation of $\mathrm{GL}_2(k)$ ($g\geq 0$ an integer). Here I consider $\mathrm{Sym}^g(k^2)$...
2
votes
1answer
122 views

How to prove that $M^G=\mathbb{F}_p[x_1\cdot v, x_1^{p\cdot(p-1)}+ v^{p-1}]$?

Let $G=Sl_2(\mathbb{F}_p)$ and $M= \mathbb{F}_p[x_1,x_2]$, where $p$ is a prime. $M$ is a $G$-module with $(A\cdot x_1, A\cdot x_2)=(x_1,x_2)\cdot A, (\forall) A \in Sl_2(\mathbb{F}_p)$. I have to ...
2
votes
1answer
232 views

Why we study Endo-Trivial Modules?

I made the exact same question on MSE some days ago and there wasn't any response whatsoever so far, so thought probably I have to ask here to get an answer. So the question goes as follows: Recently ...
6
votes
1answer
307 views

Group of order $5p^aq^b$

In Lectures by Dan Bump on Modular representation theory, Theorem 13.14 states that whenever $G$ is a non-abelian simple group of order $|G|=p^aq^br$ for distinct primes $p$,$q$, and $r$, every $r$-...
0
votes
1answer
56 views

Representations of smash products with $p$-groups

I am trying to find more generalized counterparts of some well-known results from modular group representations. My question is the following: Suppose that $H$ is a finite $p$-group acting as ...
6
votes
1answer
565 views

Finite groups with all irreducible representations one dimensional

Let $k$ be a field of characteristic $p$ and $G$ a finite group. This question might be a dulicate of this question: Which finite groups have no irreducible representations other than characters? ...
1
vote
0answers
77 views

Why is the kernel of the Brauer homomorphism a module for the normaliser?

Let $G$ be a finite group and $k$ a field. The Brauer morphism is defined as the map $$M\mapsto Br_P(M):=\frac{M^P}{\sum_Q \text{tr}^P_Q(M^Q)},$$ where $Q<P$ where $M^P$ is the points in $M$ fixed ...
4
votes
1answer
159 views

Projectives in the category of modular representations of Lie algebras

Let $\mathfrak{g}$ be a semi-simple Lie algebra (eg. $\mathfrak{g} = \mathfrak{sl}_n$), defined over an algebraically closed field $\textbf{k}$ with characteristic $p >> 0$. The center $Z(U\...
1
vote
0answers
85 views

derived invariants, perversity and modular coefficients

Let $\pi:X\rightarrow Y$ a Galois cover of finite type schemes over $\mathbb{C}$ of group $\Gamma$. Let $n$ an integer such that it is not prime with the order of $\Gamma$. Then $\pi_{*}\mathbb{Z}/n\...
6
votes
1answer
206 views

Real-valued character in Block with cyclic defect has at most two constituents modulo $p$

Let $G$ be a finite group and let $(K,R,k)$ be a $p$-modular system (large enough for $G$ etc.) and consider a block algebra $B \subseteq RG$ with cyclic defect group. My question is about the ...
2
votes
1answer
177 views

indecomposable modules restricted from $gl_n$ to $sl_n$

Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra. Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced ...
6
votes
1answer
326 views

Well-understood bases for Grothendieck groups of modular representation categories

Let $\mathfrak{g}$ be a semi-simple Lie algebra. So in characteristic $0$, the Grothendieck group of a block of category $\mathcal{O}$ is given by the classes of the Verma modules. Unlike the ...
8
votes
1answer
321 views

Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra

Definitions Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a ...
14
votes
1answer
2k views

Which finite groups have no irreducible representations other than characters?

A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...
14
votes
3answers
1k views

A question on representation of graphs

Take a complete graph $K_n$. You want to assign a vectors from $\Bbb F_2^d$ to every edge such that sum of vectors in every simple cycle does not sum to $0$ vector. The question is what is minimum $d$ ...
4
votes
0answers
254 views

Decomposition of symmetric powers of reduced regular representation modulo $p$

Let $\bar{\rho}$ denote the reduced regular representation of $\mathbb{Z}/p$ over a field of characteristic $p$. The representation $\mathrm{Sym}^k \bar{\rho}$ decomposes (for each $k$) as a sum of ...
3
votes
1answer
253 views

the number of indecomposable modules of finite groups over finite fields of a fixed dimension

I am interested in determining the the number of indecomposable modules of finite groups over finite fields of a fixed dimension. Specifically, I have the following conjecture: Conjecture. Suppose we ...
6
votes
3answers
391 views

Restriction of scalars for the adjoint representation of $SL_2(\mathbb F_q)$

Let $p$ be a prime number, $q=p^e$ a power of $p$, and $G=SL_2(\mathbb F_q)$. Let $V$ be the adjoint representation of $G$, i.e. $V$ is the 3-dimensional $\mathbb F_q$-space of of (2,2)-matrices of ...

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