Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials

We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall-Littlewood polynomials P"@l"/"@m(t) and [email protected]?s quasisymmetric Hall-Littlewood polynomials G"@c(t). More specifically, we provide:1.the G-expansions of the Hall-Littlewood polynomials P"@l(t), the monomial quasisymmetric polynomials M"@a, the quasisymmetric Schur polynomials S"@a, and the peak quasisymmetric functions K"@a; 2.an expansion of P"@l"/"@m(t) in terms of the F"@[email protected]?s. The F-expansion of P"@l"/"@m(t) is facilitated by introducing starred tableaux.

[1]  Jean-Louis Loday,et al.  Combinatorial Hopf algebras , 2008, 0810.0435.

[2]  Mike Zabrocki,et al.  A Lift of the Schur and Hall–Littlewood Bases to Non-commutative Symmetric Functions , 2014, Canadian Journal of Mathematics.

[3]  Bruce E. Sagan.  Shifted tableaux, schur Q-functions, and a conjecture of R. Stanley , 1987, J. Comb. Theory, Ser. A.

[4]  Florent Hivert,et al.  Hecke Algebras, Difference Operators, and Quasi-Symmetric Functions , 2000 .

[5]  Gregory S. Warrington,et al.  Nested Quantum Dyck Paths and ∇(sλ) , 2010 .

[6]  I. Gessel.  Multipartite P-partitions and inner products of skew Schur functions , 1983 .

[7]  S. B. Atienza-Samols,et al.  With Contributions by , 1978 .

[8]  J. Stembridge.  Enriched p-partitions , 1997 .

[9]  L. Williams,et al.  Combinatorial Hopf algebras, noncommutative Hall–Littlewood functions, and permutation tableaux , 2008, 0804.0995.

[10]  N. Loehr,et al.  Quasisymmetric expansions of Schur-function plethysms , 2012 .

[11]  J. Remmel,et al.  A combinatorial interpretation of the inverse kostka matrix , 1990 .

[12]  Mike Zabrocki,et al.  Q and Q, T-analogs of Non-commutative Symmetric Functions , 2005, Discret. Math..

[14]  J. Thibon,et al.  Hall-Littlewood functions and Kostka-Foulkes polynomials in representation theory. , 1994 .

[15]  L. Tevlin.  Noncommutative Analogs of Monomial Symmetric Functions, Cauchy Identity, and Hall Scalar Product , 2007, 0712.2201.

[17]  Alain,et al.  RIBBON TABLEAUX , HALL-LITTLEWOOD FUNCTIONS AND UNIPOTENT VARIETIES ∗ , 1997 .

[18]  Sarah Mason,et al.  Quasisymmetric Schur functions , 2011, J. Comb. Theory, Ser. A.

[19]  W. Zudilin,et al.  Dedekind's η‐function and Rogers–Ramanujan identities , 2010, 1001.1571.

[20]  Arun Ram,et al.  A combinatorial formula for Macdonald polynomials , 2008, 0803.1146.

[21]  J Haglund.  A combinatorial model for the Macdonald polynomials. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[22]  J. Stembridge.  Shifted tableaux and the projective representations of symmetric groups , 1989 .

[23]  Alain Lascoux,et al.  Noncommutative symmetric functions , 1994 .

[24]  Mark D. Haiman.  On mixed insertion, symmetry, and shifted young tableaux , 1989, J. Comb. Theory, Ser. A.

[25]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[26]  D. E. Littlewood,et al.  On Certain Symmetric Functions , 1961 .

[27]  Israel M. Gelfand,et al.  Noncommutative Symmetrical Functions , 1995 .

[28]  Joaquin O. Carbonara,et al.  A combinatorial interpretation of the inverse t-Kostka matrix , 1998, Discret. Math..

[29]  Lynne M. Butler,et al.  Subgroup Lattices and Symmetric Functions , 1994 .

[30]  Gregory S. Warrington,et al.  From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix , 2010, Eur. J. Comb..

[31]  Adriano M. Garsia,et al.  Orthogonality of Milne's polynomials and raising operators , 1992, Discret. Math..