
The space of diagonal harmonic alternants is HA_{n} = C [ E_{l} D_{n} [X] ] where D_{n} is the vandermonde determinant, E_{k} = åy_{i} ¶_{xi} and E_{l} = E_{l1} E_{l1} ¼E_{l1}. This space is naturally bigraded by \binomn2l and l(l). It is known that the dimension of HA_{n} is the Catalan number C_{n}. In fact even the bigraded dimension of HA_{n} is known as the qtCatalan number C_{n}(q,t). Yet, no explicit basis is known for this space.
We construct an explicit basis of certain graded components of HA_{n} that is valid as long as n > l.
The descent set of a sequence a_{1} a_{2} ¼a_{n} of integers is the set {i  a_{i} > a_{i+1}}. It is known that if p and s are sequences with no elements in common, then the multiset of descent sets of the shuffles of p and s depends only the descent sets of p and s. This result gives an algebra of descent sets, which is isomorphic to the algebra of quasisymmetric functions. The descent number of a sequence is the cardinality of the descent set. The descent number and several other statistics related to descents have the same shufflecompatibility property as the descent set. They correspond to certain quotients of the algebra of quasisymmetric functions, and thus to subcoalgebras of the dual coalgebra of noncommutative symmetric functions.
We study the multiplihedra, a relatively new family M of polytopes nestled between the permutahedra P and the associahedra A. The latter families were given interesting Hopf algebra structures by MalvenutoReutenauer and LodayRonco, respectively. In the work of AguiarSottile, these Hopf structures were largely explained based on geometric properties of P and A (for example, a description of their primitive elements was given in terms of the 1skeletons of the polytopes). In this talk, we define a structure on M making it a module over P and Hopf module over A. We also use its 1skeleton to exhibit the fundamental theorem of Hopf modules, giving an explicit basis of coinvariants in M. Time permitting, we indicate a whole zoo of other Hopf objects, yet to be studied, surrounding P, M, and A.
This is joint work with F. Sottile and S. Forcey.
This work is concerned with some properties of the MalvenutoReutenauer Hopf algebra of Young tableaux.
In the course of a recent study of the properties of four partial orders on Young tableaux, Taskin showed that the product of two tableaux of respective size n and m is an interval in each one of four partial orders defined on the set of tableaux of size n+m. We are interested in the relations between these intervals, with respect to the weak order on tableaux also called Young tableauhedron.
We want to show that for any quadruple (t_{1}, t_{2}, t_{3}, t_{4}) of standard Young tableaux such that t_{1} and t_{3} have the same shape l while t_{2} and t_{4} have the same shape m:
And for any couple (t_{1}, t_{2}) of standard Young tableaux: