Questions tagged [commuting-variety]
The commuting-variety tag has no usage guidance.
7
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Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )
Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...
2
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Smoothness of locus of triples $(B_1,B_2,i)$ in Nakajima's notes
In section 1.4 of Nakajima's notes on Lectures on Hilbert Schemes, it is mentioned that $(\mathbb A^2)^{[n]}$ is identified with the space of triples $\{(B_1,B_2,i)\}/GL_n$. Here $B_1,B_2$ are $n\...
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Irreducibility of linear sections of the commuting variety
The irreducibility of the commuting variety $\{(A,B) \in \mathcal{M}_{n}(\mathbb{C})^2, \ AB = BA \}$ is well-known (see for instance On Dominance and Varieties of Commuting Matrices by Gerstenhaber). ...
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3
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Variety of commuting matrices
Let $G=\operatorname{GL}(n,\mathbb{C})$ and $\mathfrak{g}=\operatorname{Mat}(n,\mathbb{C})$ and let us consider the two varieties $X,Y$ defined as $$X=\{(x,y) \in G \times G \ | \ xy=yx\} $$ and $$Y=\{...
13
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1
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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(...
- 5,594
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commuting family [closed]
I have tow matrix A & B, that B is a parametric matrix. what i can find B so that it is commuting with A?
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Request for info on the space of commuting matrices preserving a flag.
Fix a flag of subspaces V1 in V2 in V3, etc. all in Cn.
Consider the space of pairs of commuting linear transformations A and B such that:
A preserves the flag (i.e. A(Vi) is in Vi), and
B strictly ...
- 44k