r-Qsym is free over Sym

Abstract

Our main result is a proof of the Florent Hivert conjecture [F. Hivert, Local action of the symmetric group and generalizations of quasi-symmetric functions, in preparation] that the algebras of r-Quasi-Symmetric polynomials in x 1 , x 2 , . . . , x n are free modules over the ring of Symmetric polynomials. The proof rests on a theorem that reduces a wide variety of freeness results to the establishment of a single dimension bound. We are thus able to derive the Etingof-Ginzburg [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, Mosc. Math. J. 2 (2002) 555-566] Theorem on m-Quasi-Invariants and our r-QuasiSymmetric result as special cases of a single general principle. Another byproduct of the present treatment is a remarkably simple new proof of the freeness theorem for 1-Quasi-Symmetric polynomials given in [A.M. Garsia, N. Wallach, Qsym over Sym is free, J.We begin by fixing notation and recall some basic facts. We will throughout set X n = {x 1 , x 2 , . . . , x n } and denote by F[X n ] the algebra of polynomials in x 1 , x 2 , . . . , x n with coefficients in a field F of characteristic zero. If V is a graded vector space we denote by H m (V) the subspace of homogeneous elements of degree m in V. We then have the direct sum decomposition

No Feedback Card Guessing for Top to Random Shuffles

“…Diaconis et al (1992) showed that, when S = {1}, A {1} generates an n-dimensional commutative semisimple subalgebra of the group algebra, A(S n ) and A {1} has the same spectral decomposition as the transition matrix for top to random shuffles. Garsia and Wallach (2007) also proved the same result. Aldous and Diaconis (1986), Diaconis et al (1992) and recently Stark (2002) proved that n log n + cn top to random shuffles is sufficient for the deck to be random.…”

The $(1-\mathbb{E})$ -transform in combinatorial Hopf algebras

“…This problem is also an ingredient of the proof of Hivert's conjecture by Garsia and Wallach [16]. It can be solved in many different ways.…”

Generalized chromatic functions

“…The Hopf algebra of r-quasisymmetric functions is defined in [15] by Hivert. In [10], Garsia and Wallach showed that the algebra of r-quasisymmetric functions is free over symmetric functions. In the last section, we introduce the Hopf algebra of rquasisymmetric functions in noncommuting variables.…”