All Questions
Tagged with determinants rt.representation-theory
28 questions
2
votes
1
answer
144
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Determinant of finite group representation
Let $G$ be a finite group with $n$ elements. Let $\rho\colon G \to \operatorname{GL}(V)$ be a linear representation of $G$, where $V$ is a finite dimensional $\Bbb C$-vector space.
The action extends ...
- 3,432
4
votes
1
answer
237
views
A (bi)alternant formula for Wronskian
We know that there exists similarities between power functions and derivative of a function (in particular, Newton binomial formula and Leibniz rule for derivation of a product can be deduced from ...
- 275
2
votes
1
answer
214
views
A Vandermonde like determinant with exponentials
Let $n\geq m$ be non negative integers, and consider a list of $(n+m+1)$ distinct numbers (complex or real). I am interested in getting a closed form formula for the following determinant: $\det\left[\...
- 275
4
votes
0
answers
163
views
An identity for Schur polynomials
Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as
$$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-...
- 43.2k
4
votes
0
answers
78
views
Minimal set generators ideal submaximal minors
Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as:
$$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...
- 7,300
22
votes
3
answers
1k
views
On permanents and determinants of finite groups
$\DeclareMathOperator\perm{perm}$Let $G$ be a finite group. Define the determinant $\det(G)$ of $G$ as the determinant of the character table of $G$ over $\mathbb{C}$ and define the permanent $\perm(G)...
- 26.5k
2
votes
2
answers
214
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Cartan determinants of subsets
Let $n \geq 3$ be fixed.
We associate to every subset $S \subseteq \{1,...,n-1 \}$ a number, which we call Cartan determinant of $S$ (see the end of this post for a representation theoretic background)...
- 26.5k
9
votes
0
answers
123
views
Cartan determinant of stable categories
Let $A$ be a finite dimensional algebra with finitely many indecomposable non-projective modules $M_1, M_2,...,M_n$.
Let $a_{i,j}:=\dim(\underline{Hom_A}(M_j,M_i))$ (the dimension of the stable Homs ...
- 26.5k
9
votes
2
answers
425
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Matrix of cosecants appearing in equivariant index computations
In a computation of characters of certain representations of finite cyclic groups which appear as equivariant indices of Dirac operators (using the Atiyah-Bott fixed point formula, cf. [1, Theorem 8....
- 186
7
votes
1
answer
462
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On a problem for determinants associated to Cartan matrices of certain algebras
This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
- 26.5k
11
votes
2
answers
558
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Classification of algebras of finite global dimension via determinants of certain 0-1-matrices
I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...
- 26.5k
13
votes
1
answer
399
views
Is there a Giambelli identity with dual representations?
For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition.
...
- 148k
9
votes
0
answers
361
views
Which polynomials in the minors of a matrix are invariant under conjugation?
$\newcommand{\Cof}{\operatorname{cof}}$
This is a cross-post.
Let $1<k<n$ be a fixed integer. I am trying to understand "what can be said" about the $k$-degree minors of a linear map $T:V \to ...
- 6,741
1
vote
1
answer
95
views
Realising matrices as Cartan matrices
Given a matrix with natural numbers $\geq 0$ as entries and having determinant equal to one and positive diagonal entries. Is it the Cartan matrix of a finite dimensional algebra of finite global ...
- 26.5k
6
votes
0
answers
375
views
Monomial base change and the Vandermonde
Denote the falling factorials by $(x)_k=x(x-1)\cdots(x-k+1)$.
The Vandermonde determinant is given by $\det\left[x_i^{j-1}\right]_1^n=\prod_{i<j}(x_j-x_i)$.
It is well-known that in as much as ...
- 43.2k
6
votes
1
answer
208
views
Sum identities with immanants
For $\chi$ being an irreducible character of the symmetric group $S_n$ and being $M$ a complex $n\times n$-matrix, I would like to show
$$
\sum_{\sigma, \rho \in S_n} \overline{\chi(\sigma)} \chi(...
- 93
8
votes
2
answers
1k
views
Geometric interpretation of the Desnanot–Jacobi Identity
Given a square $n\times n$ matrix $M$, let $M_i^j$ denote the $(n-1)\times(n-1)$ matrix obtained from M by omitting the $i$-th row and $j$-th column of $M$.
The Desnanot–Jacobi Identity states
$$\det(...
- 81
5
votes
0
answers
620
views
Is there a method to simultaneously block-diagonalize a set of group matrices?
Assume that you are explicitly given the representation matrices of a group.
How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?
...
- 1,893
31
votes
2
answers
1k
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How to make the Capelli's identity less mysterious?
The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity
To ...
- 21.8k
11
votes
3
answers
1k
views
A class of matrix determinants between Wronskians and Vandermondes
Update: see below
Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions $...
- 4,952
3
votes
0
answers
116
views
$G$-invariant part of products of determinants of minors
Let $G = SL_n$; then for any tuple $\lambda$ such that $\sum \lambda_i = n$, define $f_\lambda(g)$ as the product of the determinants of successive minors of lengths $\lambda_i$ of $g$ (e.g. for $\...
- 4,991
8
votes
0
answers
312
views
Sum over growing Young tableaux
Let $\lambda_0,\lambda_1,\lambda_2,\lambda_3,\ldots$ be a sequence of Young diagrams, such that each successive diagram is obtained from the prior by the addition of one box (don't forget that the row ...
- 4,952
8
votes
0
answers
488
views
det(A)det(B) = det(AB+correction), Capelli identities, "factorized" representation of $\mathfrak {gl}_n$
Context: Some probably know that there are Capelli identities which state
$$det(A)det(B) = det(AB+correction)$$ for some matrices with non-commuting elements, they go back to the 19-th century, but ...
- 24.9k
6
votes
1
answer
1k
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do you know this determinant (basic commutative algebra)?
Let $\ell_1,\dots,\ell_n$ be $d+1$-variate linear forms over complex numbers in variables $X=(X_0,\dots,X_d)$. Consider the $(n-d)$-fold products
$$\ell_{i_1}(X)\ell_{i_2}(X)\dots\ell_{i_{n-d}}(X)=\...
- 14k
19
votes
4
answers
3k
views
determinant of the table of characters
I am certain that the answer to this question exists somewhere. It might be a classical exercise.
Let $G$ be a finite group. Its table of characters is a square matrix, whose rows are indexed by the ...
- 52.3k
37
votes
4
answers
3k
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What are the applications of immanants?
Definitions of determinant:
$\det(A) = \sum_{\sigma \in S_n}\operatorname{sgn} \sigma\prod_{i}a_{i, \sigma(i)}$
and permanent:
$\mathrm{per}(A) = \sum_{\sigma \in S_n}\prod_{i}a_{i, \sigma(i)}$
...
- 3,323
4
votes
1
answer
2k
views
Determinant and symmetric power
Let $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power ...
- 1,510
22
votes
3
answers
3k
views
Splitting the determinant polynomial into linear factors - a Dedekind problem
Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial
$\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq i\...
- 33.8k