Second Order Freeness and Fluctuations of Random Matrices: I. Gaussian and Wishart matrices and Cyclic Fock spaces

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of "second order freeness" and derive the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order.Comment: 46 pages, 13 figures, second revision adds explanations, figures, and reference

Automorphisms and Generalized Involution Models of Finite Complex Reflection Groups

We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either $G(r,p,n)$ with $\gcd(p,n)=1$; $G(r,p,2)$ with $r/p$ odd; or $G_{23}$, the Coxeter group of type $H_3$. We additionally provide explicit formulas for all automorphisms of $G(r,p,n)$, and construct new Gelfand models for the groups $G(r,p,n)$ with $\gcd(p,n)=1$.Comment: 29 page

The Robinson-Schensted Correspondence and A2A_2-web Bases

We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to $[n,n,n]$: the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and the {\em left cell basis} for the left cell construction of Kazhdan and Lusztig. In the case of $[n,n]$, the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the images of these bases under classical maps: the {\em Robinson-Schensted algorithm} between permutations and Young tableaux and {\em Khovanov-Kuperberg's bijection} between Young tableaux and reduced webs. One main result uses Vogan's generalized $\tau$-invariant to uncover a close structural relationship between the web basis and the left cell basis. Intuitively, generalized $\tau$-invariants refine the data of the inversion set of a permutation. We define generalized $\tau$-invariants intrinsically for Kazhdan-Lusztig left cell basis elements and for webs. We then show that the generalized $\tau$-invariant is preserved by these classical maps. Thus, our result allows one to interpret Khovanov-Kuperberg's bijection as an analogue of the Robinson-Schensted correspondence. Despite all of this, our second main result proves that the reduced web and left cell bases are inequivalent; that is, these bijections are not $S_{3n}$-equivariant maps.Comment: 34 pages, 23 figures, minor corrections and revisions in version