Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra \(\mathsf {P}_k(n)\) and the symmetric group \(\mathsf {S}_n\) are in Schur–Weyl duality on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\), so there is a surjection \(\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is an isomorphism when \(n \ge 2k\). We prove a dimension formula for the irreducible modules of the centralizer algebra \(\mathsf {Z}_k(n)\) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible \(\mathsf {S}_n\)-modules in \(\mathsf {M}_n^{\otimes k}\). Our dimension expressions hold for any \(n \ge 1\) and \(k\ge 0\). Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on \(\mathsf {M}_n^{\otimes k}\) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of \(\mathsf {S}_n\).     

Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time

We consider a branching random walk on \({\mathbb {R}}\) with a stationary and ergodic environment \(\xi =(\xi _n)\) indexed by time \(n\in {\mathbb {N}}\). Let \(Z_n\) be the counting measure of particles of generation n and \(\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)\) be its Laplace transform. We show the \(L^p\) convergence rate and the uniform convergence of the martingale \(\tilde{Z}_n(t)/{\mathbb {E}}[\tilde{Z}_n(t)|\xi ]\), and establish a moderate deviation principle for the measures \(Z_n\).     

Ranks of ideals in inverse semigroups of difunctional binary relations

The set \(\mathcal {D}_n\) of all difunctional relations on an n element set is an inverse semigroup under a variation of the usual composition operation. We solve an open problem of Kudryavtseva and Maltcev (Publ Math Debrecen 78(2):253–282, 2011), which asks: What is the rank (smallest size of a generating set) of \(\mathcal {D}_n\)? Specifically, we show that the rank of \(\mathcal {D}_n\) is \(B(n)+n\), where B(n) is the nth Bell number. We also give the rank of an arbitrary ideal of \(\mathcal {D}_n\). Although \(\mathcal {D}_n\) bears many similarities with families such as the full transformation semigroups and symmetric inverse semigroups (all contain the symmetric group and have a chain of \(\mathscr {J}\)-classes), we note that the fast growth of \({\text {rank}}(\mathcal {D}_n)\) as a function of n is a property not shared with these other families.     

On finitely generated modules over quasi-Euclidean rings

Let R be a unital commutative ring, and let M be an R-module that is generated by k elements but not less. Let \(\text {E}_n(R)\) be the subgroup of \(\text {GL}_n(R)\) generated by the elementary matrices. In this paper we study the action of \(\text {E}_n(R)\) by matrix multiplication on the set \(\text {Um}_n(M)\) of unimodular rows of M of length \(n \ge k\). Assuming R is moreover Noetherian and quasi-Euclidean, e.g., R is a direct product of finitely many Euclidean rings, we show that this action is transitive if \(n > k\). We also prove that \(\text {Um}_k(M) /\text {E}_k(R)\) is equipotent with the unit group of \(R/\mathfrak {a}_1\) where \(\mathfrak {a}_1\) is the first invariant factor of M. These results encompass the well-known classification of Nielsen non-equivalent generating tuples in finitely generated Abelian groups.     

On symplectic semifield spreads of $$\hbox {PG}(5, q^2)$$PG(5, q2), q even

We prove that the only symplectic semifield spreads of \(\hbox {PG}(5,q^2)\), \(q\ge 2^{14}\) even, whose associated semifield has center containing \({\mathbb F}_q\), is the Desarguesian spread, by proving that the only \({\mathbb F}_q\)-linear set of rank 6 disjoint from the secant variety of the Veronese surface of \(\hbox {PG}(5,q^2)\) is a plane with three points of the Veronese surface of \(\hbox {PG}(5,q^6){\setminus } \hbox {PG}(5,q^2)\).     

On eigenvalues of Seidel matrices and Haemers’ conjecture

For a graph G, let S(G) be the Seidel matrix of G and \({\theta }_1(G),\ldots ,{\theta }_n(G)\) be the eigenvalues of S(G). The Seidel energy of G is defined as \(|{\theta }_1(G)|+\cdots +|{\theta }_n(G)|\). Willem Haemers conjectured that the Seidel energy of any graph with n vertices is at least \(2n-2\), the Seidel energy of the complete graph with n vertices. Motivated by this conjecture, we prove that for any \(\alpha \) with \(0<\alpha <2,|{\theta }_1(G)|^\alpha +\cdots +|{\theta }_n(G)|^\alpha \geqslant (n-1)^\alpha +n-1\) if and only if \(|\hbox {det}\,S(G)|\geqslant n-1\). This, in particular, implies the Haemers’ conjecture for all graphs G with \(|\hbox {det}\,S(G)|\geqslant n-1\). A computation on the fraction of graphs with \(|\hbox {det}\,S(G)|<n-1\) is reported. Motivated by that, we conjecture that almost all graphs G of order n satisfy \(|\hbox {det}\,S(G)|\geqslant n-1\). In connection with this conjecture, we note that almost all graphs of order n have a Seidel energy of order \(\Theta (n^{3/2})\). Finally, we prove that self-complementary graphs G of order \(n\equiv 1\pmod 4\) have \(\det S(G)=0\).     

A family of discrete semi-classical orthogonal polynomials of class one

In this work, we solve the system of Laguerre–Freud equations for the recurrence coefficients \(\beta _n\), \(\gamma _{n+1} , n \ge 0\) of the \(D_{w}\)-semi-classical orthogonal polynomials sequences of class one in the case when \(\beta _{0}=-t_{0}\), \(\beta _{n+1}=t_{n}-t_{n+1}\) and \(\gamma _{n+1}=-t_{n}^{2}\) with \(t_{n}\ne 0\;n\ge 0\), where \(D_w\) is the divided difference operator. There are essentially four canonical families.     

Semigroups of rectangular matrices under a sandwich operation

Let \({\mathcal {M}}_{mn}={\mathcal {M}}_{mn}({\mathbb {F}})\) denote the set of all \(m\times n\) matrices over a field \({\mathbb {F}}\), and fix some \(n\times m\) matrix \(A\in {\mathcal {M}}_{nm}\). An associative operation \(\star \) may be defined on \({\mathcal {M}}_{mn}\) by \(X\star Y=XAY\) for all \(X,Y\in {\mathcal {M}}_{mn}\), and the resulting sandwich semigroup is denoted \({\mathcal {M}}_{mn}^A={\mathcal {M}}_{mn}^A({\mathbb {F}})\). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study \({\mathcal {M}}_{mn}^A\) as well as its subsemigroups \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\) and \({\mathcal {E}}_{mn}^A\) (consisting of all regular elements and products of idempotents, respectively), and the ideals of \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\). Among other results, we characterise the regular elements; determine Green’s relations and preorders; calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider; and classify the isomorphisms between finite sandwich semigroups \({\mathcal {M}}_{mn}^A({\mathbb {F}}_1)\) and \({\mathcal {M}}_{kl}^B({\mathbb {F}}_2)\). Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of partial semigroups related to Ehresmann-style “arrows only” categories; we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variants \({\mathcal {M}}_n^A\) of the full linear monoid \({\mathcal {M}}_n\) (in the case \(m=n\)), and to certain semigroups of linear transformations of restricted range or kernel (in the case that \(\hbox {rank}(A)\) is equal to one of m, n).     

Lifting Maps from the Symmetrized Polydisc in Small Dimensions

The spectral unit ball \(\Omega _n\) is the set of all \(n\times n\) matrices M with spectral radius less than 1. Let \(\pi (M) \in \mathbb {C}^n\) stand for the coefficients of the characteristic polynomial of a matrix M (up to signs), i.e. the elementary symmetric functions of its eigenvalues. The symmetrized polydisc is \({{\mathbb {G}}}_n:=\pi (\Omega _n)\). When investigating Nevanlinna–Pick problems for maps from the disk to the spectral ball, it is often useful to project the map to the symmetrized polydisc (for instance to obtain continuity results for the Lempert function): if \(\Phi \in {\mathrm {Hol}}(\mathbb {D}, \Omega _n)\), then \(\pi \circ \Phi \in {\mathrm {Hol}}(\mathbb {D}, {{\mathbb {G}}}_n)\). Given a map \(\varphi \in {\mathrm {Hol}}(\mathbb {D}, {{\mathbb {G}}}_n)\), we are looking for necessary and sufficient conditions for this map to “lift through given matrices”, i.e. find \(\Phi \) as above so that \(\pi \circ \Phi = \varphi \) and \(\Phi (\alpha _j) = A_j\), \(1\le j \le N\). A natural necessary condition is \(\varphi (\alpha _j)=\pi (A_j)\), \(1\le j \le N\). When the matrices \(A_j\) are derogatory (i.e. do not admit a cyclic vector) new necessary conditions appear, involving derivatives of \(\varphi \) at the points \(\alpha _j\). We prove that those conditions are necessary and sufficient for a local lifting. We give a formula which performs the global lifting in small dimensions (\(n \le 5\)), and a counter-example to show that the formula fails in dimensions 6 and above.     

Subconvexity for sup-norms of cusp forms on $$\mathrm{PGL}(n)$$

Let F be an \(L^2\)-normalized Hecke Maaß cusp form for \(\Gamma _0(N) \subseteq {\mathrm{SL}}_{n}({\mathbb {Z}})\) with Laplace eigenvalue \(\lambda _F\). If \(\Omega \) is a compact subset of \(\Gamma _0(N)\backslash {\mathrm{PGL}}_n/\mathrm{PO}_{n}\), we show the bound \(\Vert F|_{\Omega }\Vert _{\infty } \ll _{ \Omega } N^{\varepsilon } \lambda _F^{n(n-1)/8 - \delta }\) for some constant \(\delta = \delta _n> 0\) depending only on n.     

Some new expansions for certain truncated <Emphasis Type="Italic">q</Emphasis>-series

We introduce the notion of \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials, where \(\mathcal {R}_{\mu }\) is the degree raising shift operator for the sequence of Laguerre polynomials of parameter \(\mu \). Then we show that the Laguerre polynomials \(L^{(\mu )}_n(x), \ \mu \ne -m, \ m\ge 0\), are the only \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials.     

Eldan’s Stochastic Localization and Tubular Neighborhoods of Complex-Analytic Sets

Let \(f: \mathbb {C}^n \rightarrow \mathbb {C}^k\) be a holomorphic function and set \(Z = f^{-1}(0)\). Assume that Z is non-empty. We prove that for any \(r > 0\),

$$\begin{aligned} \gamma _n(Z + r) \ge \gamma _n(E + r), \end{aligned}$$

where \(Z + r\) is the Euclidean r-neighborhood of Z; \(\gamma _n\) is the standard Gaussian measure in \(\mathbb {C}^n\), and \(E \subseteq \mathbb {C}^n\) is an \((n-k)\)-dimensional, affine, complex subspace whose distance from the origin is the same as the distance of Z from the origin.

     

Euclidean Distance Between Haar Orthogonal and Gaussian Matrices

In this work, we study a version of the general question of how well a Haar-distributed orthogonal matrix can be approximated by a random Gaussian matrix. Here, we consider a Gaussian random matrix \(Y_n\) of order n and apply to it the Gram–Schmidt orthonormalization procedure by columns to obtain a Haar-distributed orthogonal matrix \(U_n\). If \(F_i^m\) denotes the vector formed by the first m-coordinates of the ith row of \(Y_n-\sqrt{n}U_n\) and \(\alpha \,=\,\frac{m}{n}\), our main result shows that the Euclidean norm of \(F_i^m\) converges exponentially fast to \(\sqrt{ \big (2-\frac{4}{3} \frac{(1-(1 -\alpha )^{3/2})}{\alpha }\big )m}\), up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm \(\epsilon _n(m)\,=\,\sup _{1\le i \le n, 1\le j \le m} |y_{i,j}- \sqrt{n}u_{i,j}|\) and we find a coupling that improves by a factor \(\sqrt{2}\) the recently proved best known upper bound on \(\epsilon _n(m)\). Our main result also has applications in Quantum Information Theory.     

On the Second Boundary Value Problem for a Class of Fully Nonlinear Equations

We proved the existence of convex solution to a class of fully nonlinear elliptic equations with second boundary condition on uniformly convex domains in \(\mathbb {R}^{n}\), and then applied it to solve a boundary value problem for minimal Lagrangian graphs in the pseudo-Euclidean space \(\mathbb {R}^{2n}_n\).     

Anderson Polymer in a Fractional Brownian Environment: Asymptotic Behavior of the Partition Function

We consider the Anderson polymer partition function

$$\begin{aligned} u(t):=\mathbb {E}^X\left[ e^{\int _0^t \mathrm {d}B^{X(s)}_s}\right] \,, \end{aligned}$$

where \(\{B^{x}_t\,;\, t\ge 0\}_{x\in \mathbb {Z}^d}\) is a family of independent fractional Brownian motions all with Hurst parameter \(H\in (0,1)\), and \(\{X(t)\}_{t\in \mathbb {R}^{\ge 0}}\) is a continuous-time simple symmetric random walk on \(\mathbb {Z}^d\) with jump rate \(\kappa \) and started from the origin. \(\mathbb {E}^X\) is the expectation with respect to this random walk. We prove that when \(H\le 1/2\), the function u(t) almost surely grows asymptotically like \(e^{\lambda t}\), where \(\lambda >0\) is a deterministic number. More precisely, we show that as t approaches \(+\infty \), the expression \(\{\frac{1}{t}\log u(t)\}_{t\in \mathbb {R}^{>0}}\) converges both almost surely and in the \(\hbox {L}^1\) sense to some positive deterministic number \(\lambda \). For \(H>1/2\), we first show that \(\lim _{t\rightarrow \infty } \frac{1}{t}\log u(t)\) exists both almost surely and in the \(\hbox {L}^1\) sense and equals a strictly positive deterministic number (possibly \(+\infty \)); hence, almost surely u(t) grows asymptotically at least like \(e^{\alpha t}\) for some deterministic constant \(\alpha >0\). On the other hand, we also show that almost surely and in the \(\hbox {L}^1\) sense, \(\limsup _{t\rightarrow \infty } \frac{1}{t\sqrt{\log t}}\log u(t)\) is a deterministic finite real number (possibly zero), hence proving that almost surely u(t) grows asymptotically at most like \(e^{\beta t\sqrt{\log t}}\) for some deterministic positive constant \(\beta \). Finally, for \(H>1/2\) when \(\mathbb {Z}^d\) is replaced by a circle endowed with a Hölder continuous covariance function, we show that \(\limsup _{t\rightarrow \infty } \frac{1}{t}\log u(t)\) is a deterministic finite positive real number, hence proving that almost surely u(t) grows asymptotically at most like \(e^{c t}\) for some deterministic positive constant c.

     

On kernels and nuclei of rank metric codes

For each rank metric code \(\mathcal {C}\subseteq \mathbb {K}^{m\times n}\), we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When \(\mathcal {C}\) is \(\mathbb {K}\)-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When \(\mathbb {K}\) is a finite field \(\mathbb {F}_q\) and \(\mathcal {C}\) is a maximum rank distance code with minimum distance \(d<\min \{m,n\}\) or \(\gcd (m,n)=1\), the kernel of the associated translation structure is proved to be \(\mathbb {F}_q\). Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over \(\mathbb {F}_q\) must be a finite field; its right nucleus also has to be a finite field under the condition \(\max \{d,m-d+2\} \geqslant \left\lfloor \frac{n}{2} \right\rfloor +1\). Let \(\mathcal {D}\) be the DHO-set associated with a bilinear dimensional dual hyperoval over \(\mathbb {F}_2\). The set \(\mathcal {D}\) gives rise to a linear rank metric code, and we show that its kernel and right nucleus are isomorphic to \(\mathbb {F}_2\). Also, its middle nucleus must be a finite field containing \(\mathbb {F}_q\). Moreover, we also consider the kernel and the nuclei of \(\mathcal {D}^k\) where k is a Knuth operation.     

On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$, their $$\mathbb {Z}_4$$Z4 images,and new codes

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).     

Characterization of some derivations on von Neumann algebras via left centralizers

Let \(\mathfrak {M}\) be a von Neumann algebra, and let \(\mathfrak {T}:\mathfrak {M} \rightarrow \mathfrak {M}\) be a bounded linear map satisfying \(\mathfrak {T}(P^{2}) = \mathfrak {T}(P)P + \Psi (P,P)\) for each projection P of \(\mathfrak {M}\), where \(\Psi :\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is a bi-linear map. If \(\Psi \) is a bounded l-semi Hochschild 2-cocycle, then \(\mathfrak {T}\) is a left centralizer associated with \(\Psi \). By applying this conclusion, we offer a characterization of left \(\sigma \)-centralizers, generalized derivations and generalized \(\sigma \)-derivations on von Neumann algebras. Moreover, it is proved that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every bi-\(\sigma \)-derivation \(D:\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.     

Self-Extensional Three-Valued Paraconsistent Logics

A logic \(\mathbf{L}\) is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \(\mathbf{L}\)-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, there is exactly one self-extensional three-valued paraconsistent logic in the language of \(\{\lnot ,\wedge ,\vee \}\) for which \(\vee \) is a disjunction, and \(\wedge \) is a conjunction. We also investigate the main properties of this logic, determine the expressive power of its language (in the three-valued context), and provide a cut-free Gentzen-type proof system for it.