Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables

We analyze the structure of the algebra N of symmetric polynomials in non-commuting variables in so far as it relates to its commutative counterpart. Using the "place-action" of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of N analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.Comment: 14 page

Quasisymmetric harmonics of the exterior algebra

We study the ring of quasisymmetric polynomials in $n$ anticommuting (fermionic) variables. Let $R_n$ denote the polynomials in $n$ anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in $R_n$ form a commutative sub-algebra of $R_n$. (2) There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by $$\text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,$$ where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$. (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot conditionComment: 17 pages, changed list of authors, minor corrections to pape

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (nonself-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by radical relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C*-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the weak-operator closures of these algebras as well.Comment: 46 pages. Final version, to appear in Advances in Mathematic

Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system $(W,S)$ we construct a certain noncommutative algebra, so-called {\it bracket algebra}, together with a familiy of commuting elements, so-called {\it Dunkl elements.} Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the group $W.$ We prove this conjecture for classical Coxeter groups and $I_2(m)$. We define a ``quantization'' and a multiparameter deformation of our construction and show that for Lie groups of classical type and $G_2,$ the algebra generated by Dunkl elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define {\it quantum Bruhat representation} of the corresponding bracket algebra. We study in more detail relations and structure of $B_n$-, $D_n$- and $G_2$-bracket algebras, and as an application, discover {\it Pieri type formula} in the $B_n$-bracket algebra. As a corollary, we obtain Pieri type formula for multiplication of arbitrary $B_n$-Schubert classes by some special ones. Our Pieri type formula is a generalization of Pieri's formulas obtained by A. Lascoux and M.-P. Sch\"utzenberger for flag varieties of type $A.$ We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements which describes ``noncommutative differential geometry on a finite Coxeter group'' in a sense of S. Majid

Oddification of the cohomology of type A Springer varieties

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q=-1. This allows us to define graded modules over the Hecke algebra at q=-1 that are `odd' analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient of the skew polynomial ring by the left ideal of nonconstant odd symmetric functions. The top degree component of the odd cohomology of Springer varieties is identified with the corresponding Specht module of the Hecke algebra at q=-1.Comment: 21 pages, 2 eps file