Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras

We present a family of rank symmetric diamond-colored distributive lattices that are naturally related to the Fibonacci sequence and certain of its generalizations. These lattices re-interpret and unify descriptions of some un- or differently-colored lattices found variously in the literature. We demonstrate that our symmetric Fibonaccian lattices naturally realize certain (often reducible) representations of the special linear Lie algebras, with weight basis vectors realized as lattice elements and Lie algebra generators acting along the order diagram edges of each lattice. We present evidence that each such weight basis is, in a certain sense, uniquely associated with its lattice. We provide new descriptions of the lattice cardinalities and rank generating functions and offer several conjectures/open problems. Throughout, we make connections with integer sequences from the OEIS.Comment: 18 page

A Survey of Alternating Permutations

This survey of alternating permutations and Euler numbers includes refinements of Euler numbers, other occurrences of Euler numbers, longest alternating subsequences, umbral enumeration of classes of alternating permutations, and the cd-index of the symmetric group.Comment: 32 pages, 7 figure

The peak algebra and the Hecke-Clifford algebras at q=0q=0

Using the formalism of noncommutative symmetric functions, we derive the basic theory of the peak algebra of symmetric groups and of its graded Hopf dual. Our main result is to provide a representation theoretical interpretation of the peak algebra and its graded dual as Grothendieck rings of the tower of Hecke-Clifford algebras at $q=0$.Comment: Final version, 17 pages, LaTex, 1 PDF figure, graphic

Tensor algebras, words, and invariants of polynomials in non-commutative variables

Consider a vector space V for which we specify a basis, then the tensor algebra T(V) corresponds to the non-commutative polynomials expressed in that basis. If V has an S_n module structure (more generally, for a finite group) then identifying the invariants of the non-commutative polynomials corresponds to calculating the multiplicity of the trivial representation in the repeated Kronecker product of the Frobenius image of the module V. We consider a general method of arriving at a combinatorial interpretation for the Kronecker coefficients by embedding the representation ring within a group algebra. This is joint work with Anouk Bergeron-Brlek and Christophe Hohlweg