Let $F$ be an algebraically closed field of characteristic $0$ and let $\operatorname{sp}(2l,F)$ be the rank $l$ symplectic algebra of all $2l\times 2l$ matrices $x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$ over $F$, where $A^{t}$ is the transpose of $A$ and $B,C$ are symmetric matrices of order $l$. The commuting graph $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ of $\operatorname{sp}(2l,F)$ is a graph whose vertex set consists of all nonzero elements in $\operatorname{sp}(2l,F)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$. We prove that the diameter of $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ is $4$ when $l>2$.

Let $S$ be a surface, $G$ a simply connected classical group, and $G^{\prime }$ the associated adjoint form of the group. We show that the moduli spaces of framed local systems ${\mathcal{X}}_{G^{\prime },S}$ and ${\mathcal{A}}_{G,S}$, which were constructed by Fock and Goncharov [‘Moduli spaces of local systems and higher Teichmuller theory’, Publ. Math. Inst. Hautes Études Sci.103 (2006), 1–212], have the structure of cluster varieties, and thus together form a cluster ensemble. This simplifies some of the proofs in that paper, and also allows one to quantize higher Teichmuller space, which was previously only possible when $G$ was of type $A$.

Let $K$ be a field that admits a cyclic Galois extension of degree $n\geq 2$. The symmetric group $S_{n}$ acts on $K^{n}$ by permutation of coordinates. Given a subgroup $G$ of $S_{n}$ and $u\in K^{n}$, let $V_{G}(u)$ be the $K$-vector space spanned by the orbit of $u$ under the action of $G$. In this paper we show that, for a special family of groups $G$ of affine type, the dimension of $V_{G}(u)$ can be computed via the greatest common divisor of certain polynomials in $K[x]$. We present some applications of our results to the cases $K=\mathbb{Q}$ and $K$ finite.

We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.

We generalize the Cohen–Lenstra heuristics over function fields to étale group schemes $G$ (with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg–Venkatesh–Westerland, we make progress towards the proof of these heuristics. Moreover, by keeping track of the image of the Weil-pairing as an element of $\wedge ^{2}G(1)$, we formulate more refined heuristics which nicely explain the deviation from the usual Cohen–Lenstra heuristics for abelian $\ell$-groups in cases where $\ell \mid q-1$; the nature of this failure was suggested already in the works of Malle, Garton, Ellenberg–Venkatesh–Westerland, and others. On the purely large random matrix side, we provide a natural model which has the correct moments, and we conjecture that these moments uniquely determine a limiting probability measure.

We consider a class of sample covariance matrices of the form Q = TXX*T*, where X = (xij) is an M×N rectangular matrix consisting of independent and identically distributed entries, and T is a deterministic matrix such that T*T is diagonal. Assuming that M is comparable to N, we prove that the distribution of the components of the right singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments of xij coincide with the Gaussian random variables. For the right singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments of xij match those of the Gaussian random variables. Similar results hold for the left singular vectors if we further assume that T is diagonal.

We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0 and $A\in \mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan–Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in \mathfrak{s}$, $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$) such that $A=S+N$.

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let $M_{N}$ be a deterministic $N\times N$ matrix, and let $G_{N}$ be a complex Ginibre matrix. We consider the matrix ${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$, where $\unicode[STIX]{x1D6FE}>1/2$. With $L_{N}$ the empirical measure of eigenvalues of ${\mathcal{M}}_{N}$, we provide a general deterministic equivalence theorem that ties $L_{N}$ to the singular values of $z-M_{N}$, with $z\in \mathbb{C}$. We then compute the limit of $L_{N}$ when $M_{N}$ is an upper-triangular Toeplitz matrix of finite symbol: if $M_{N}=\sum _{i=0}^{\mathfrak{d}}a_{i}J^{i}$ where $\mathfrak{d}$ is fixed, $a_{i}\in \mathbb{C}$ are deterministic scalars and $J$ is the nilpotent matrix $J(i,j)=\mathbf{1}_{j=i+1}$, then $L_{N}$ converges, as $N\rightarrow \infty$, to the law of $\sum _{i=0}^{\mathfrak{d}}a_{i}U^{i}$ where $U$ is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when $\mathfrak{d}=1$, also of independent and identically distributed entries on the diagonals in $M_{N}$.

We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives an example where the partial transpose produces freeness at the operator level. Finally, we investigate the case of real Wishart matrices.

We introduce the concept of a $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$-Rota–Baxter operator, as a twisted version of a Rota–Baxter operator of weight zero. We show how to obtain a certain $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$-Rota–Baxter operator from a solution of the associative (Bi)Hom-Yang–Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.

Three families of examples are given of sets of $(0,1)$-matrices whose pairwise products form a basis for the underlying full matrix algebra. In the first two families, the elements have rank at most two and some of the products can have multiple entries. In the third example, the matrices have equal rank $\!\sqrt{n}$ and all of the pairwise products are single-entried $(0,1)$-matrices.

Let $G$ be an infinite graph on countably many vertices and let $\unicode[STIX]{x1D6EC}$ be a closed, infinite set of real numbers. We establish the existence of an unbounded self-adjoint operator whose graph is $G$ and whose spectrum is $\unicode[STIX]{x1D6EC}$.

In this paper, we study the relative perturbation bounds for joint eigenvalues of commuting tuples of normal $n\times n$ matrices. Some Hoffman–Wielandt-type relative perturbation bounds are proved using the Clifford algebra technique. We also extend a result for diagonalisable matrices which improves a relative perturbation bound for single matrices.

We aim to characterise those transformations on the set of density operators (which are the mathematical representatives of the states in quantum information theory) that preserve a so-called generalised entropy of one fixed convex combination of operators. The characterisation strengthens a recent result of Karder and Petek where the preservation of the same quantity was assumed for all convex combinations.

For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups by analysing the passage of significant properties from Cu(A) to Cu(A)⊗Cu Cu(). We describe the effect of the natural map Cu(A) → Cu(A)⊗Cu Cu() in the order of Cu(A), and show that if A has real rank 0 and no elementary subquotients, Cu(A)⊗Cu Cu() enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, non-elementary, real rank 0 and stable rank 1 situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with is inert at the level of the Cuntz semigroup.

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the $\text{Gl}_{n}(\mathbb{K})$ action on the $n\times n$ matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one–one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in $M_{n}(\mathbb{K})$. We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams.

We prove that every endomorphism of an infinite-dimensional vector space over a field splits into the sum of four idempotents and into the sum of four square-zero endomorphisms, a result that is optimal in general.

New inequalities relating the norm $n(X)$ and the numerical radius $w(X)$ of invertible bounded linear Hilbert space operators were announced by Hosseini and Omidvar [‘Some inequalities for the numerical radius for Hilbert space operators’, Bull. Aust. Math. Soc.94 (2016), 489–496]. For example, they asserted that $w(AB)\leq$$2w(A)w(B)$ for invertible bounded linear Hilbert space operators $A$ and $B$. We identify implicit hypotheses used in their discovery. The inequalities and their proofs can be made good by adding the extra hypotheses which take the form $n(X^{-1})=n(X)^{-1}$. We give counterexamples in the absence of such additional hypotheses. Finally, we show that these hypotheses yield even stronger conclusions, for example, $w(AB)=w(A)w(B)$.

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame, which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.