All Questions
12 questions
2
votes
0
answers
117
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A very specific quotient of a determinantal variety
I'm interesting in knowing whether a certain variety defined by maximal minors is irreducible. The specific construction is as follows: let $n \geq 2$ and let $R = \mathbb{C}[a_1,b_1,c_1,d_1,e_1,f_1,...
- 185
2
votes
0
answers
148
views
Nilpotent polynomial matrices over $F_q$ - polynomial count variety ? ( Nilpotent cone for Hitchin-Gaudin like integrable system)
Context: Number of nilpotent $n\times n $ matrices over $F_q$ is $q^{n(n-1)}$ classical result due to Ph.Hall, M.Gerstenhaber (see very nice exposition by T.Leinster at n-cat-cafe/arxiv) which have ...
- 24.9k
0
votes
1
answer
134
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Existence of cyclic subspace decompositions for pairs of commuting matrices
Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute.
For $v\in V$, ...
- 16.2k
6
votes
1
answer
260
views
Vanishing linear combinations of minors
Let $V$ be the set of $k$ by $n$ matrices ($k<n$) with entries in $\mathbb{C}$, and let $\mathbb{C}[V]$ denote the set of polynomial functions on $V$. For any subset $I \subseteq [n] = \{1,2,\dotsc,...
- 7,300
1
vote
1
answer
209
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Primary decomposition of huge ideals using M2/Singular
I used to ask similar questions in other communities, but so far never received any feedback.
Given four Hermitian $n\times n$ matrices $A_1,A_2,B_1,B_2$ together with the constraints $[A_i,B_j]=0$, I ...
- 71
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:
$$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...
- 7,300
4
votes
1
answer
127
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On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion
Consider the polynomial ring $R=\mathbb C[x,y]$.
Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...
- 2,784
2
votes
1
answer
244
views
radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices
Consider the polynomial ring $R=\mathbb C[x_1,x_2,...,x_{16}]$, and set
$$X=\begin{pmatrix} x_1 &x_2&x_3 &x_4\\ x_5&x_6& x_7&x_8\\x_9&x_{10}&x_{11}&x_{12}\\x_{13}&...
- 5,039
3
votes
1
answer
402
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the pfaffian-adjugate and its counterparts for matrices odd size
Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$.
If $n$ is even, then $\det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...
- 2,232
5
votes
1
answer
786
views
Vector Spaces of Symmetric Matrices of Low Rank
Let $K$ be a field, if necessary algebraic closed or of characteristic zero. Let $k$ be a positive integer. I am interested in linear subspaces $M \subseteq \textrm{Sym}_n(K)$, where $\textrm{Sym}_n(K)...
1
vote
1
answer
324
views
Simultaneous triangularizability over a commutative ring
Let $R$ be a commutative ring with unity and $A,B\in M_n(R)$ satisfying the property
(*) All elements of the two-side ideal, in $M_n(R)$, generated by $AB-BA$, are nilpotent.
McCoy showed that, if $...
- 4,111
2
votes
2
answers
798
views
Determinantal rings are Cohen-Macaulay
Consider a $n\times n$ matrix $M$ with entries in $R=\mathbb{C}[x_1,\dots,x_n]$. Let $I$ be the ideal of $(n-1)\times(n-1)$ minors of $M$. Is $\mathcal{O}_{\mathbb{C}^n}/I$ Cohen-Macaulay?If not, ...
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