Polynomial identities for the Jordan algebra of a degenerate symmetric bilinear form

Let _Jn$J_n$ be the Jordan algebra of a degenerate symmetric bilinear form. In the first section we classify all possible G  -gradings on _Jn$J_n$ where G   is any group, while in the second part we restrict our attention to a degenerate symmetric bilinear form of rank n−1$n-1$, where n is the dimension of the vector space V   defining _Jn$J_n$. We prove that in this case the algebra _Jn$J_n$ is PI-equivalent to the Jordan algebra of a nondegenerate bilinear form.     

The Zassenhaus filtration,Massey products,and representations of profinite groups

We consider the p  -Zassenhaus filtration (_Gn)$(G_n)$ of a profinite group G  . Suppose that G=S/N$G=S/N$ for a free profinite group S and a normal subgroup N of S   contained in _Sn$S_n$. Under a cohomological assumption on the n-fold Massey products (which holds, e.g., if G has p  -cohomological dimension ≤ 1), we prove that _Gn+1$G_{n+1}$ is the intersection of all kernels of upper-triangular unipotent (n+1)$(n+1)$-dimensional representations of G   over _Fp${\mathbb{F}}_p$. This extends earlier results by Miná?, Spira, and the author on the structure of absolute Galois groups of fields.     

Representation theory and homological stability

We introduce the idea of representation stability   (and several variations) for a sequence of representations _Vn$V_n$ of groups _Gn$G_n$. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical theorems of homological stability to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood–Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (the (n+1)ⁿ⁻¹${(n+1)}^{n-1}$ conjecture). The majority of this paper is devoted to exposing this phenomenon through examples. In doing this we obtain applications, theorems and conjectures.     

On matrices over an arbitrary semiring and their generalized inverses

In this paper, we consider matrices with entries from a semiring S. We first discuss some generalized inverses of rectangular and square matrices. We establish necessary and sufficient conditions for the existence of the Moore–Penrose inverse of a regular matrix. For an m×n$m\times n$ matrix A  , an n×m$n\times m$ matrix P and a square matrix Q of order m, we present necessary and sufficient conditions for the existence of the group inverse of QAP   with the additional property that P(QAP)^#Q$P{(QAP)}^{\mathrm{\#}}Q$ is a {1,2}$\{1,2\}$ inverse of A  . The matrix product used here is the usual matrix multiplication. The result provides a method for generating elements in the set of {1,2}$\{1,2\}$ inverses of an m×n$m\times n$ matrix A starting from an initial {1} inverse of A  . We also establish a criterion for the existence of the group inverse of a regular square matrix. We then consider a semiring structure (_Mm×n(S),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$ made up of m×n$m\times n$ matrices with the addition defined entry-wise and the multiplication defined as in the case of the Hadamard product of complex matrices. In the semiring (_Mm×n(S),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$, we present criteria for the existence of the Drazin inverse and the Moore–Penrose inverse of an m×n$m\times n$ matrix. When S is commutative, we show that the Hadamard product preserves the Hermitian property, and provide a Schur-type product theorem for the product A°(CC^?)$A°(C{C}^{?})$ of a positive semidefinite n×n$n\times n$ matrix A   and an n×n$n\times n$ matrix C.     

Functions of perturbed n-tuples of commuting self-adjoint operators

Let (_A1,…,_An)$(A_1,\dots ,A_n)$ and (_B1,…,_Bn)$(B_1,\dots ,B_n)$ be n-tuples of commuting self-adjoint operators on Hilbert space. For functions f   on Rⁿ${\mathbb{R}}^n$ satisfying certain conditions, we obtain sharp estimates of the operator norms (or norms in operator ideals) of f(_A1,…,_An)−f(_B1,…,_Bn)$f(A_1,\dots ,A_n)-f(B_1,\dots ,B_n)$ in terms of the corresponding norms of _Aj−_Bj$A_j-B_j$, 1?j?n$1?j?n$. We obtain analogs of earlier results on estimates for functions of perturbed self-adjoint and normal operators. It turns out that for n?3$n?3$, the methods that were used for self-adjoint and normal operators do not work. We propose a new method that works for arbitrary n  . We also get sharp estimates for quasicommutators f(_A1,…,_An)R−Rf(_B1,…,_Bn)$f(A_1,\dots ,A_n)R-Rf(B_1,\dots ,B_n)$ in terms of norms of _AjR−R_Bj$A_{j}R-R{B}_j$, 1?j?n$1?j?n$, for a bounded linear operator R.     

Proof of the Andrews–Dyson–Rhoades conjecture on the spt-crank

The spt-crank of a vector partition, or an S  -partition, was introduced by Andrews, Garvan and Liang. Let _NS(m,n)$N_S(m,n)$ denote the net number of S-partitions of n with spt-crank m, that is, the number of S  -partitions (_π1,_π2,_π3)$({\pi }_1,{\pi }_2,{\pi }_3)$ of n with spt-crank m   such that the length of _π1${\pi }_1$ is odd minus the number of S  -partitions (_π1,_π2,_π3)$({\pi }_1,{\pi }_2,{\pi }_3)$ of n with spt-crank m   such that the length of _π1${\pi }_1$ is even. Andrews, Dyson and Rhoades conjectured that _{NS(m,n)}m${\{N_S(m,n)\}}_m$ is unimodal for any n  , and they showed that this conjecture is equivalent to an inequality between the rank and crank of ordinary partitions. They obtained an asymptotic formula for the difference between the rank and crank of ordinary partitions, which implies _NS(m,n)≥_NS(m+1,n)$N_S(m,n)\ge N_S(m+1,n)$ for sufficiently large n and fixed m. In this paper, we introduce a representation of an ordinary partition, called the m-Durfee rectangle symbol, which is a rectangular generalization of the Durfee symbol introduced by Andrews. We give a proof of the conjecture of Andrews, Dyson and Rhoades. We also show that this conjecture implies an inequality between the positive rank and crank moments obtained by Andrews, Chan and Kim.     

A note on p-completely bounded homomorphisms of the Figà-Talamanca–Herz algebras

For a locally compact group G   and 1<p<∞$1<p<\infty$ let _Ap(G)$A_p(G)$ be the Figà-Talamanca–Herz algebras, which include in particular the Fourier algebra of G  , A(G)$A(G)$ (p=2$p=2$). It is shown that for any amenable group H  , a proper affine map α:Y⊂H→G$\alpha :Y\subset H\to G$ induces a p  -completely contractive algebra homomorphism _?α:_Ap(G)→_Ap(H)${?}_{\alpha }:A_p(G)\to A_p(H)$ by setting _?α(u)=u°α${?}_{\alpha }(u)=u°\alpha$ on Y   and _?α(u)=0${?}_{\alpha }(u)=0$ off of Y. Moreover, we show that if both G and H are amenable then any p  -completely contractive algebra homomorphism ?:_Ap(G)→_Ap(H)$?:A_p(G)\to A_p(H)$ is of this form. These results are the analogs in the context of the Figà-Talamanca–Herz algebras of the ones in the Fourier algebra setting (p=2$p=2$) initiated by the author and continued with N. Spronk, which in turn generalize results of P.J. Cohen and B. Host from abelian group algebra setting.     

Higher genus mapping class group invariants from factorizable Hopf algebras

Lyubashenko?s construction associates representations of mapping class groups _Mapg:n${\mathrm{Map}}_{g:n}$ of Riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category. We consider this construction as applied to the category of bimodules over a finite-dimensional factorizable ribbon Hopf algebra H  . For any such Hopf algebra we find an invariant of _Mapg:n${\mathrm{Map}}_{g:n}$ for all values of g and n  . More generally, we obtain such invariants for any pair (H,ω)$(H,\omega )$, where ω is a ribbon automorphism of H.