Small maximal spaces of non-invertible matrices

Permutation representations on invertible matrices

We discuss permutation representations which are obtained by the natural action of $S_n \times S_n$ on some special sets of invertible matrices, defined by simple combinatorial attributes. We decompose these representations into irreducibles. The multiplicities involved have a nice combinatorial interpretation. We also generalize known results on asymptotic behavior of the conjugacy representation of $S_n$.

平面网络与可逆矩阵的多重 Horn 问题

Multiple Horn problems for planar networks and invertible matrices

本文研究平面网络多重 Horn 问题，提出迹等式、菱形不等式和四面体等式作为必要条件，并猜想其在网络足够大时也充分；同时证明乘法问题在大缩放参数下 Duistermaat-Heckman 测度集中于八面体递推轨迹附近。

Sign-coherence of c-vectors and maximal green sequences for acyclic sign-skew-symmetric matrices

In this paper we construct an unfolding for $c-$vectors of acyclic sign-skew symmetric matrices and we also prove that the sign-coherence property holds for acyclic sign-skew-symmetric matrices. Then we prove that every acyclic sign-skew-symmetric matrix admits a maximal green sequence.

Proportion of cyclic matrices in maximal reducible matrix algebras

Let ${\rm M}(V)={\rm M}(n,\mathbb{F}_q)$ denote the algebra of $n\times n$ matrices over $\mathbb{F}_q$, and let ${\rm M}(V)_U$ denote the (maximal reducible) subalgebra that normalizes a given $r$-dimensional subspace $U$ of $V=\mathbb{F}_q^n$ where $0<r<n$. We prove that the density of non-cyclic matrices in ${\rm M}(V)_U$ is at least $q^{-2}\left(1+c_1q^{-1}\right)$, and at most $q^{-2}\left(1+c_2q^{-1}\right)$, where $c_1$ and $c_2$ are constants independent of $n,r$, and $q$. The constants $c_1=-\frac43$ and $c_2=\frac{35}3$ suffice.

Uniformly column sign-coherence and the existence of maximal green sequences

In this paper, we prove that each matrix in $M_{m\times n}(\mathbb Z_{\geq0})$ is uniformly column sign-coherent with respect to any $n\times n$ skew-symmetrizable integer matrix. Using such matrices, we introduce the definition of irreducible skew-symmatrizable matrix. Based on this, the existence of a maximal green sequence for a skew-symmetrizable matrices is reduced to the existence of a maximal green sequence for irreducible skew-symmetrizable matrices.

The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations

For an arbitrary associative unital ring $R$, let $J_1$ and $J_2$ be the following noncommutative birational partly defined involutions on the set $M_3(R)$ of $3\times 3$ matrices over $R$: $J_1(M)=M^{-1}$ (the usual matrix inverse) and $J_2(M)_{jk}=(M_{kj})^{-1}\,$ (the transpose of the Hadamard inverse). We prove the following surprising conjecture by Kontsevich saying that $(J_2\circ J_1)^3$ is the identity map modulo the ${\rm Diag}_{L} \times \rm{Diag}_R$ action $(D_1,D_2)(M)=D_1^{-1}MD_2$ of pairs of invertible diagonal matrices. That is, we show that for each $M$ in the domain where $(J…