All Questions
Tagged with rt.representation-theory ra.rings-and-algebras
424 questions
2
votes
0
answers
63
views
Finding non-commutative finite-dimensional "hypersurface" algebras
Fix a field $K$.
Call a non-commutative polynomial $f(x_i)$ whose monomial terms are all of degree at least 2 in the variables $x_i$ magic if the finite dimensional $K$-algebra $A_{f,n}:=K<x_i>/(...
- 26.5k
2
votes
0
answers
382
views
Is there a roadmap to learning representation theory of finite group over finite field?
I've been wanted to learn some basic theories of the (non-semisimple) representation of the finite group over a finite field.
I have been guessing that the materials might be contained in the books on ...
- 1,013
2
votes
1
answer
317
views
English translation of Emmy Noether's Hyperkomplexe Grössen und Darstellungstheorie
I'm wondering if anybody knows where one can find an English translation of Emmy Noether's classical paper E. NOETHER, Hyperkomplexe Grössen und Darstellungstheorie, Math. Zeit. 30(1929), 641–692 ?...
- 38.6k
3
votes
2
answers
449
views
Is there any simple formula for the character of $S_{n}$ represented by the set of $k$-tuples of $\{1,2,...,n\}$?
I'm interested in the representation theory of symmetric groups.
I'm now trying to search for the formula for the characters of $\Omega^{k}$, the set of $k$-tuple of elements of $\Omega$ a set of $n$ ...
- 1,013
5
votes
0
answers
161
views
Representation theory terminology question
For a paper I'm writing, I need a term for a representation-theoretic concept that I'm sure someone has thought of before, so I thought I'd ask here rather than just make something up.
Let $G$ be a ...
- 44.8k
2
votes
0
answers
74
views
Explicit example of prime ideal not an intersection of maximal ideals, in universal enveloping algebra
Let $A$ be a $\mathbb k$-algebra. If $A$ is affine commutative, by Nullstellensatz, then every prime ideal of $A$ is an intersection of maximal ideals.
To justify the notion of being primitive in ...
- 485
4
votes
1
answer
324
views
What is the $p$-regular partition corresponding to the sign representation of $S_{n}$ over a field of characteristic $p$?
I'm now interested in the modular representation of symmetric groups.
It is well-known that for a fixed prime $p$, there is a bijection between the irreducible representations of $S_{n}$ over a field ...
- 1,013
2
votes
0
answers
104
views
$G$-module representations of a profinite quiver
I have a profinite directed graph $\Gamma$, i.e., I can think of $\Gamma$ as the inverse limit of a directed system of finite directed graphs under inclusion. To each vertex of the graph a $G$-module ...
- 21
2
votes
0
answers
90
views
Regular conjugacy classes and irreducible representations in the infinite, projective case
Let $k$ be an algebraically closed field and $G$ a (not necessarily finite) group. Let $\alpha\colon G\times G\to k^*$ be a multiplier, meaning that
$\alpha(s,t)\alpha(st,r)=\alpha(s,tr)\alpha(t,r)$ ...
- 1,903
4
votes
0
answers
80
views
Finding all nice ideals for quiver algebras
Let $Q$ be a finite, connected and acyclic quiver which is simply-laced.
Let $k$ be a field and $kQ$ the path algebra of $Q$ over $k$.
Recall that an ideal $I$ of $kQ$ is called admissible if it is ...
- 26.5k
2
votes
0
answers
81
views
The centralizer and normalizer of products of (SU(n) $\times$ SU(p) $\times$ …) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}
$Consider the special unitary group $\SU(n)$ and the unitary group $\U(m)$.
Below I specify a specfic way to embed $...
- 10.5k
2
votes
0
answers
111
views
The centralizer and normalizer of products of (Spin(n) $\times \dots$) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$
Consider the spin group $\Spin(n)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $(\Spin(...
- 10.5k
3
votes
1
answer
355
views
The normalizer of SU(n) in U(m)?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $...
- 10.5k
6
votes
1
answer
215
views
What is the smallest group for which Broué's abelian defect group conjecture has not yet been verified?
Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $k:=\overline{\mathbb{F}_p}$.
Let $b$ be a $p$-block of $kG$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...
- 1,755
1
vote
1
answer
275
views
The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$?
$\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ I can show that
$$
\U(2^{N-1})\supset \Spin(2N)
$$
when $2N > 4$ or a positive integer $N > 2$, so $\Spin(2N)$ can be embedded in $\U(2^...
- 10.5k
10
votes
1
answer
307
views
Rings where all indecomposable projective modules are finitely generated
Let $X$ be the class of (unital, associative and not necessarily commutative) rings $R$ where every indecomposable projective $R$-module is finitely generated.
Question 1: Is there a nice equivalent ...
- 26.5k
6
votes
1
answer
294
views
Rickard's strengthening of Broué's abelian defect group conjecture and the lifting of some equivalences up to splendid derived equivalences
Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $K:=\overline{\mathbb{F}_p}$.
Let $b$ be a $p$-block of $G$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...
- 1,755
6
votes
1
answer
137
views
On the finiteness of an Auslander-Reiten component
I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by
CLAUDIA CHAIO AND SHIPING LIU. This is Theorem 2.7:
And this is part of it's proof, in which the direction (2) $\Rightarrow $ ...
- 251
-1
votes
1
answer
143
views
infinite left degrees
I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by
CLAUDIA CHAIO AND SHIPING LIU. This is a part of the paper:
Definition: Let $f: X \rightarrow Y$ be an irreducible morphism ...
- 251
0
votes
2
answers
283
views
Motivation and reference for Brauer algebras
I am looking for a good reference and motivation for Brauer monoid and Brauer algebras. Kindly help me with some suggestions. Thanks.
- 141
1
vote
1
answer
201
views
About composition factors [closed]
I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by
CLAUDIA CHAIO AND SHIPING LIU. This is part of the proof of Lemma 2.3
$A$ is assumed to be an Artin algebra and mod(A) the ...
- 251
9
votes
0
answers
366
views
A characterisation of symmetric algebras using Hochschild (co)homology
A finite dimensional (connected if needed) $K$-algebra $A$ over a field $K$ is called symmetric when $A \cong Hom_K(A,K)$ as $A$-bimodules. Symmetric algebras are Frobenius algebras and include for ...
- 26.5k
1
vote
1
answer
437
views
Question on simple modules and projective covers
I have the following question:
Let $A$ be an Artin algebra. Let $S_1$ and $S_2$ be simple modules in $\text{mod}(A)$ and let $P(S_1)$ be the projective cover of $S_1$. Let $f: P(S_1) \rightarrow S_2$ ...
- 251
3
votes
2
answers
202
views
Question on injective hulls
How can I show the following:
Let $f: M \rightarrow N$ be a morphism in $\text{mod}(A)$, where $A$ is an Artin algebra. Suppose $f \neq 0$. Then there exists a simple module $S$ with its injective ...
- 251
5
votes
1
answer
618
views
Uniqueness of infinite direct sum decomposition
A module $M$ over a ring $R$ is called semisimple if it admits a direct sum decomposition into simple modules. If $M$ admits a finite decomposition $M=\bigoplus_{i=1}^n S_i$ into simple $R$-modules $...
user78294
1
vote
0
answers
89
views
Is there any English reference for the paper 'Darstellungstheorie von Schur-Algebren' written by Fredrich Roesler?
Now I'm reading the paper of Friedrich Roesler on the representation theory of Schur-Rings with the title 'Darstellungstheorie von Schur-Algebren' (Math Z 1972).
My goal is to understand algebraic ...
- 1,013
2
votes
0
answers
75
views
Is the regular representation of a fusion ring a direct sum of all its irreducible representations?
For groups, the regular representation contains each irrep with multiplicity equal to the irrep's dimension.
For (not necessarily commutative) fusion rings, is there any analogous statement for the ...
- 437
3
votes
0
answers
107
views
Do Frobenius algebras have a lattice basis and what lattices do appear?
Let $K$ be for simplicity be the field with two or three elements (or alternatively we could restrict to ideals containing only the field elements $-1$ or $1$ as coefficients).
A (commutative) ...
- 26.5k
8
votes
0
answers
251
views
When does a semisimple $\mathbb{C}$-algebra come from a group?
Let $\mathcal{A}$ be a semisimple $\mathbb{C}$-algebra. By the Artin-Wedderburn theorem, it is isomorphic to a direct product of matrix algebras:
$$ \mathcal{A} = \prod_{i=1}^m M_{n_i}(\mathbb{C})$$
...
- 297
3
votes
0
answers
100
views
Tensor product of modules over twisted differential operators
Let $R$ be an algebra over complex numbers. Let $N$ be a module over $R$. We can define the algebra $D(N)$ of differential operators $N \rightarrow N$ using Grothendieck’s approach as follows: we ...
- 163
4
votes
1
answer
228
views
When is the enveloping algebra finitely generated over its center as a module?
Let $g$ be a Lie algebra with enveloping algebra $U(g)$ and $Z(g)$ the center of $U(g)$.
Question 1: When is $Z(g)$ noetherian?
Question 2: When is $U(g)$ a finitely generated $Z(g)$-module? Is this ...
- 26.5k
5
votes
1
answer
179
views
Outer automorphism group of posets
Let $X$ be a finite poset (we can assume it is connected) and $A_K(X)$ the incidence algebra of $X$ over a field $K$.
The following result is well known, see for example corollary 7.3.7 in the book &...
- 26.5k
3
votes
1
answer
270
views
Thin representations for quiver algebras
A representation $M$ of a quiver is called thin when $M$ has a dimension vector consisting only of 0 or 1 entries.
When $A=kQ$ is a path algebra for a tree $Q$, then there is the nice result that ...
- 26.5k
2
votes
1
answer
197
views
Top and bottom composition factors of $M$ are isomorphic
Let $k$ be a field and $N$ a finite group. Let $M$ be a projective indecomposable $kN$-module. Since the algebra $kN$ is symmetric, it follows that the top and bottom composition factors of $M$ are ...
- 51
2
votes
1
answer
160
views
MAGMA-question concerning the transformation of a $kG$ -module $M$ into a right ideal of the group algebra
Let $G$ be a finite group and $k$ be a finite field of characteristic $p>0$ such that $p\mid |G|$.
Let $M$ be a $kG$-module which has an embedding $M\hookrightarrow kG^{reg}$ into the regular $kG$-...
- 1,755
6
votes
1
answer
281
views
An identity for Ext for rings
Let $A$ be a two-sided noetherian ring (which we should assume to be Gorenstein first so that everything is well defined, otherwise it is only well defined up to a conjecture, which states that every ...
- 26.5k
6
votes
2
answers
235
views
Is there a CAS that can solve a given system of equations in a finite group algebra $kG$?
Let $k$ be a finite field with char$(k)=p>0$. Let $G$ be a finite group.
Consider the group algebra $kG$.
I would like to solve a given system of equations in $kG$.
Question:
Is there a computer ...
- 1,755
0
votes
0
answers
76
views
Isomorphism problem for enveloping algebras
Let A and B be finite-dimensional algebras over a field k. We denote the enveloping algebra of A by A^e, which is the tensor product (over k) of the algebra A and its opposite algebra. Suppose A^e and ...
3
votes
0
answers
66
views
Isomorphism of algebras depending on the field
Let $X$ be a finite set of primes (where 0 is also a prime, but 1 is not!).
Are there two quiver algebra $A_1=KQ/I_1$ and $A_2=KQ/I_2$ such that the coefficeints of the generators of the ideals $I_1$ ...
- 26.5k
1
vote
0
answers
151
views
Parameter of Brauer algebra
Let $O(V)$= the set of orthogonal transformation from the vector space $V$ to $V$ where $\dim V=n$. We know that the centralizer algebra of $O(V)$ on the tensor space $V^{\otimes{f}}$ is Brauer ...
- 179
1
vote
0
answers
107
views
Reference request concerning splitting fields for groups that are related to special symmetric groups
Denote the symmetric group of order $n!$ by $S_n$. Let $H:=S_p$ for an odd prime $p$.
Every finite field $k$ is a splitting field $(^*)$ for $kH$, in particular $k:=\mathbb{F}_p$.
Questions:
Is $k:=\...
- 311
7
votes
1
answer
232
views
Looking for citable reference for a well-known fact about tensor product of finite dimensional algebras over an algebraically closed field
Let $K$ be an algebraically closed field and let $A$ and $B$ be finite dimensional algebras over $K$. Let $e_1,\ldots, e_n$ be orthogonal primitive idempotents of $A$ summing to $1$ and $f_1,\ldots, ...
- 38.6k
5
votes
1
answer
829
views
Rigid monoidal and closed monoidal categories
I am trying to understand the relationship between rigid monoidal categories and closed monoidal
categories. First every rigid monoidal category is closed, with an adjoint to the functor $X \otimes -$ ...
- 1,144
3
votes
2
answers
279
views
Searching for theorems characterizing when $O_p(G)$ is trivial / non-trivial
Let $G$ be a finite group. Let $p$ be a prime.
Let $O_p(G)$ be the $p$-core of $G$.
Are there any theorems known saying something like
$O_p(G)$ is trivial, if and only if ... and
$O_p(G)$ is non-...
- 237
3
votes
1
answer
95
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 $\mu$ ...
- 1,755
3
votes
0
answers
61
views
On grades of torsion modules in noetherian rings
Let $A$ be a (not necessarily commutative) two-sided noetherian ring with minimal injective coresolution $(I_i)$ of the regular module $A$ as a right module.
Say that $A$ has dominant dimension $n$ in ...
- 26.5k
15
votes
4
answers
869
views
What is known about ordinary character values at involutions?
Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$.
Let $x$ be an involution in $G$.
I'd like to ask the following
Question 1:
...
- 1,755
4
votes
0
answers
153
views
The Jacobson radical as a bimodule
Let $A$ be a finite dimensional algebra with Jacobson radical $J$.
Question 1: In case $A$ is a Nakayama algebra with a linear quiver corresponding to a Dyck path $D$ (via its Auslander-Reiten ...
- 26.5k
4
votes
3
answers
344
views
Coinvariants of tensor products of Hopf algebras
Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way.
The axioms of Hopf algebras imply that
$$
G^{\operatorname{coinv}(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \...
4
votes
1
answer
159
views
Effect of extending scalars on maps of modules
Let $k$ be a field and $R$ be a $k$-algebra. Let $M$ and $N$ be left $R$-modules. Finally, let $\ell$ be a field extension of $k$. We thus have an $\ell$-algebra $\ell \otimes R$, and both $\ell \...
- 43