All Questions
424 questions
38
votes
0
answers
1k
views
Groups whose complex irreducible representations are finite dimensional
By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting.
It is easy ...
- 38.6k
33
votes
2
answers
2k
views
What do cluster algebras tell us about Grassmannians?
One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
- 1,690
28
votes
4
answers
5k
views
When are modules and representations not the same thing?
I've been trying for a while to get a real concrete handle on the relationship between representations and modules. To frame the question, I'll put here the standard situation I have in mind:
A ring $...
- 1,872
27
votes
5
answers
3k
views
Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?
Let $R$ be a finitely generated ring with identity, $M_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? This should be an elementary ...
- 1,373
27
votes
3
answers
3k
views
Is there a 'nice' interpretation of virtual representations?
Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those ...
- 5,191
24
votes
6
answers
7k
views
Introduction to W-Algebras/Why W-algebras?
Does anyone know of an introduction and motivation for W-algebras?
Edit: Okay, sorry I try to add some more background. W algebras occur, for example when you study nilpotent orbits: Take a nice ...
- 13.2k
23
votes
1
answer
1k
views
Codes, lattices, vertex operator algebras
At the end of "Notes on Chapter 1" in the Preface to the Third Edition of Sphere packings, lattices and groups, Conway and Sloane write the following:
Finally, we cannot resist calling attention to ...
- 2,150
21
votes
3
answers
2k
views
Is there a "categorical" description of Grothendieck's algebra of differential operators?
First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due ...
- 54.6k
19
votes
2
answers
851
views
The discriminant of the Okada algebra
The Okada algebra $\mathfrak{O}_n$ over a field $K$ has generators
$E_1,\dots,E_{n-1}$ and relations $E_i^2=x_iE_i$,
$E_{i+1}E_iE_{i+1}=y_i E_{i+1}$, and $E_iE_j=E_jE_i$ for $|i-j|\geq
2$, where $x_i,...
- 50.8k
18
votes
2
answers
1k
views
Does base extension reflect the property of being isomorphic?
Let L/K be a (separable?) field extension, let A be a finite dimensional algebra over K, and let M and N be two A-modules. Let $A' = L \otimes_K A$ be the algebra given by extension of scalars, and ...
- 28.1k
17
votes
1
answer
419
views
Freeness of tensor product
Let $G$ be a finite group. Is $\mathbb{Z}G\otimes_{Z(\mathbb{Z}G)}\mathbb{Z}G$ free as a $\mathbb{Z}$-module, where $Z$ denotes the centre?
- 355
17
votes
0
answers
704
views
When is the determinant an $8$-th power?
I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...
- 7,300
15
votes
3
answers
3k
views
Which is the correct universal enveloping algebra in positive characteristic?
This is an extension of this question about symmetric algebras in positive characteristic. The title is also a bit tongue-in-cheek, as I am sure that there are multiple "correct" answers.
Let $\...
- 54.6k
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
15
votes
1
answer
638
views
What is the centralizer of a Young subgroup of $S_n$?
In their celebrated paper "A new approach to the representation theory of the symmetric group. II", Okounkov and Vershik prove that $Z(n-1,1)$, the centralizer of $\mathbb{C}[S_{n-1}]$ in $\...
- 492
15
votes
1
answer
758
views
Swan K-theory of Z/4
Given a finite group $G$ and a commutative ring $R$, define the Swan $K$-theory $K_0(G, R)$ to be the Grothendieck group of the category finitely generated projective $R$-modules with $G$-action (with ...
- 25.6k
14
votes
4
answers
2k
views
A finite dimensional algebra associated to the symmetric group
Let $S_n$ be the finite group given as $n \times n$ permutation matrices.
Define for a given field $K$ the algebra $B_n$ as the subalgebra of $M_n(K)$ generated by all permutation matrices of $S_n$. (...
- 26.5k
14
votes
2
answers
741
views
Is a "separable" algebra over a field finite-dimensional?
Let $k$ be a field and $A$ a unital associative (possibly non-commutative) $k$-algebra, and let $A^e$ denote the enveloping algebra of $A$, namely, $A^e = A \otimes_k A^{op}$.
It seems that there are ...
- 149
14
votes
2
answers
1k
views
Name for algebra and its tensor products
Is there a name for the algebra (and its tensor products) given by generators $M_{j}$, $j \in \mathbb{Z}_{n}$ under the conditions $$M_{j} = (1 - M_{j-1})(1-M_{j+1})$$ where $j=1\implies j-1=n$ and $j=...
- 13.9k
13
votes
4
answers
3k
views
What is a "block" in an abelian category?
In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which ...
- 52.9k
13
votes
1
answer
5k
views
What are tame and wild hereditary algebras?
What are tame and wild hereditary algebras?
Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. ...
- 2,992
13
votes
4
answers
2k
views
Geometric interpretation of Universal enveloping algebras
Given a complex Lie algebra $\mathfrak g$, we can form its universal enveloping algebra and interpret it as a noncommutative space.
Is this perspective useful? What does this space "look like"?
How ...
- 13.2k
13
votes
3
answers
2k
views
Classification of commutative Frobenius algebras
Are there attempts to classify commutative finite dimensional Frobenius algebras? They appear often in mathematics, such as in algebraic geometry and the famous category equivalence between ...
- 26.5k
13
votes
1
answer
698
views
Counting representations of $k[x,y]$ when $k$ is finite
$\newcommand{\GFq}{\mathbf F_q}$
Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial $p_n(...
- 5,654
13
votes
1
answer
669
views
Is a "smooth" finite-dimensional algebra separable modulo its radical?
Let $k$ be a field, and let us write the "unadorned" tensor $\otimes$ in place of $\otimes_k$. For a unital finite-dimensional $k$-algebra $A$, let $A^e = A \otimes A^{op}$ denote the enveloping ...
- 5,407
13
votes
1
answer
598
views
Is algebra: ac=ca, bd = db , ad - da = cb - bc ("Manin matrix algebra") - a Koszul algebra?
Question: Consider quadratic algebra with four generators $a,b,c,d,d$ and three relations $ac=ca,bd = db, ad-da = cb - bc$ . Is it a Koszul algebra ? (i.e. Koszul complex is resolution of ground field ...
- 24.8k
13
votes
1
answer
1k
views
Why not _co_free modules?
Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and ...
- 54.6k
13
votes
0
answers
355
views
Analog of Haar element in an algebra
In a Hopf algebra $H $ (over some field $ k $), there is the notion of a Haar element $ h \in H$. This is an element of the algebra which has the property that if $ V $ is a representation of $ H $, ...
- 4,612
12
votes
1
answer
922
views
Does this algebra have finite global dimension ? (Human vs computer)
Usually computers can calculate the global dimension of a finite dimensional quiver algebra much faster than humans. But in this case a high end computer (calculating for 3 weeks) was not able to ...
- 26.5k
12
votes
1
answer
564
views
Representations of degenerate Clifford algebras
Given a real finite-dimensional vector space $V$ with a symmetric bilinear form $b$, we define the Clifford algebra $Cl(V,b)$ as the quotient of the tensor algebra $\bigotimes V$ by the two sided ...
- 33.3k
12
votes
0
answers
605
views
Given an algebra, can it be realized as a block of a Hopf algebra?
During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in k$...
- 2,450
11
votes
2
answers
604
views
Temperley-Lieb algebras for other Weyl groups?
The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen ...
- 27.8k
11
votes
1
answer
329
views
Are there three non-commutative polynomials in three variables with finite dimensional quotient?
$\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $K$ be a field and $K\la x,y,z\ra$ the non-commutative polynomial ring in 3 variables.
Question 1: Are there three (fewer is probably not possible?!...
- 26.5k
11
votes
1
answer
159
views
Are all indecomposable $\mathbb{Z}_+$-modules over the character ring of a group, character rings of a subgroup?
A $\mathbb Z_+$-algebra is an algebra $A$ over $\mathbb C$ with given basis $\{v_i\}$ such that
$$v_iv_j=\sum_k n_{ijk}v_k,\hspace{10mm}n_{ijk}\in\mathbb Z_{\geq0}.$$
An example of such an object is ...
- 301
11
votes
0
answers
436
views
A rather strange algebra
Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...
- 3,318
11
votes
0
answers
708
views
What happens to simple modules under Ringel duality?
If $A$ is a quasi-hereditary algebra then its Ringel dual $A'=End_A(T)$ is the endomorphism algebra of a (minimal) full tilting module $T$ for $A$. The algebra $A'$ is again quasi-hereditary, although ...
- 578
11
votes
0
answers
286
views
What do Multilinear Forms tell us about Representations?
The last few days I have been calculating whether certain group representations are real, complex, or quaternionic. It is well-known that the type of the representation corresponds to what type of ...
- 5,191
10
votes
2
answers
2k
views
Geometric interpretation of matrix minors
i was recently interested in geometric interpretation of various notions showing up in linear algebra because in most cases linear algebra with geometry courses focus too much on linear algebra not ...
- 221
10
votes
1
answer
272
views
Plane partitions as irreducible representations
The irreducible representations of the symmetric group algebras $A_n=KS_n$ over a the complex numbers (or any field of characteristic 0) $K$ satisfy the following properties:
The irreducible ...
- 26.5k
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
10
votes
2
answers
2k
views
Isomorphism between a filtered vector space and its associated graded
$\DeclareMathOperator\gr{gr}$Let $ V $ be a vector space with a decreasing filtration
$$
V = F_0 V \supseteq F_1 V \supseteq F_2 V \supseteq\dotsb .$$
We define the associated graded of $ V $ to be $$ ...
- 4,612
10
votes
0
answers
275
views
Are local finite dimensional Hopf algebras symmetric?
Recall that a finite dimensional algebra $A$ over a field $K$is called a Frobenius algebra in case $A \cong D(A)$ as right modules, where $D(A) \cong Hom_K(A,K)$. In case $A \cong D(A)$ as bimodules, ...
- 26.5k
9
votes
2
answers
2k
views
A ring for which the category of left and right modules are distinct
What is an example of a ring $R$ for which the abelian category of left $R$-modules is not isomorphic to the category of right $R$-modules?
- 435
9
votes
2
answers
2k
views
What are examples of cogenerators in R-mod?
Fill in the blank, please :)
Let $\mathcal C$ be a complete and cocomplete abelian category. A generator in $\mathcal C$ is an object $X \in \mathcal C$ such that every object $Y \in \mathcal C$ ...
- 54.6k
9
votes
3
answers
987
views
Octonionic Unitary Group?
Hi all.
I was wondering if anyone has any references on work related to the Octonionic Unitary group. I would imagine that such a group would be generated by Octonionic skew-Hermitian matrices (at ...
- 177
9
votes
3
answers
1k
views
Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
9
votes
1
answer
472
views
Why do we want $p$-permutation modules in splendid equivalences?
First Rickard (in Splendid Equivalences: Derived Categories and Permutation Modules ) and then Rouquier (Block theory via stable and Rickard equivalences, Appendix A.1) define splendid equivalences ...
9
votes
2
answers
626
views
Smallest faithful matrix representation of the exterior algebra
Let $R = \Lambda \mathbb{C}^n$ be the exterior algebra on $\mathbb{C}^n$ for some positive integer $n$. It is an associative (graded-commutative) algebra of $\mathbb{C}$-dimension $2^n$.
Suppose we ...
- 2,851
9
votes
2
answers
815
views
Strategies for proving a category is Noetherian?
Let $C$ be a small linear category over a commutative ring $R$. A representation of $C$ is an $R$-linear functor $C \to \mathrm{Mod}(R)$. For example, for each $c\in C$, there is a representation $[...
- 54.6k
9
votes
2
answers
3k
views
Jacobson radical = intersection of all maximal two-sided ideals
I'm embarassed to ask this question, but the literature on noncommutative rings seems to give this a berth as if it was absolutely trivial and not worth discussing, and I can't prove it, so all I can ...
- 33.8k