On Urn Models, Non-commutativity and Operator Normal Forms

Non-commutativity is ubiquitous in mathematical modeling of reality and in many cases same algebraic structures are implemented in different situations. Here we consider the canonical commutation relation of quantum theory and discuss a simple urn model of the latter. It is shown that enumeration of urn histories provides a faithful realization of the Heisenberg-Weyl algebra. Drawing on this analogy we demonstrate how the operator normal forms facilitate counting of histories via generating functions, which in turn yields an intuitive combinatorial picture of the ordering procedure itself.Comment: 7 pages, 2 figure

The toric ideal of a graphic matroid is generated by quadrics

Describing minimal generating sets of toric ideals is a well-studied and difficult problem. Neil White conjectured in 1980 that the toric ideal associated to a matroid is generated by quadrics corresponding to single element symmetric exchanges. We give a combinatorial proof of White's conjecture for graphic matroids.Comment: 19 pages, 4 figure

Cyclage, catabolism, and the affine Hecke algebra

We identify a subalgebra \pH_n of the extended affine Hecke algebra \eH_n of type A. The subalgebra \pH_n is a \u-analogue of the monoid algebra of \S_n \ltimes \ZZ_{\geq 0}^n and inherits a canonical basis from that of \eH_n. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n, which we term \emph{positive affine tableaux} (PAT). We then exhibit a cellular subquotient \R_{1^n} of \pH_n that is a \u-analogue of the ring of coinvariants \CC[y_1,...,y_n]/(e_1,...,e_n) with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element \pi \in \pH_n corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that \R_{1^n} has cellular quotients \R_\lambda that are \u-analogues of the Garsia-Procesi modules R_\lambda with left cells labeled by (a PAT version of) the \lambda-catabolizable tableaux. We give a conjectural description of a cellular filtration of \pH_n, the subquotients of which are isomorphic to dual versions of \R_\lambda under the perfect pairing on \R_{1^n}. We conjecture how this filtration relates to the combinatorics of the cells of \eH_n worked out by Shi, Lusztig, and Xi. We also conjecture that the k-atoms of Lascoux, Lapointe, and Morse and the R-catabolizable tableaux of Shimozono and Weyman have cellular counterparts in \pH_n. We extend the idea of atom copies of Lascoux, Lapoint, and Morse to positive affine tableaux and give descriptions, mostly conjectural, of some of these copies in terms of catabolizability.Comment: 58 pages, youngtab.sty included for tableau

Combinatorial Route to Algebra: The Art of Composition & Decomposition

We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication laws, thereby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the concepts of composition and decomposition with the emphasis on combinatorial origin of the ensuing algebraic constructions.Comment: 20 pages, 6 figure

Kronecker coefficients for one hook shape

We give a positive combinatorial formula for the Kronecker coefficient g_{lambda mu(d) nu} for any partitions lambda, nu of n and hook shape mu(d) := (n-d,1^d). Our main tool is Haiman's \emph{mixed insertion}. This is a generalization of Schensted insertion to \emph{colored words}, words in the alphabet of barred letters \bar{1},\bar{2},... and unbarred letters 1,2,.... We define the set of \emph{colored Yamanouchi tableaux of content lambda and total color d} (CYT_{lambda, d}) to be the set of mixed insertion tableaux of colored words w with exactly d barred letters and such that w^{blft} is a Yamanouchi word of content lambda, where w^{blft} is the ordinary word formed from w by shuffling its barred letters to the left and then removing their bars. We prove that g_{lambda mu(d) nu} is equal to the number of CYT_{lambda, d} of shape nu with unbarred southwest corner.Comment: 37 pages, 3 figure