Skew quasisymmetric Schur functions and noncommutative Schur functions

Abstract Recently a new basis for the Hopf algebra of quasisymmetric functions QSym, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset L C that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions NSym. This basis of NSym is dual to the basis of quasisymmetric Schur functions and its elements are the pre-image of the Schur functions under the forgetful map χ : NSym → Sym . We prove that the multiplicative structure constants of the noncommutative Schur functions, equivalently the coefficients of the skew quasisymmetric Schur functions when expanded in the quasisymmetric Schur basis, are nonnegative integers, satisfying a Littlewood–Richardson rule analogue that reduces to the classical Littlewood–Richardson rule under χ. As an application we show that the morphism of algebras from the algebra of Poirier–Reutenauer to Sym factors through NSym. We also extend the definition of Schur functions in noncommuting variables of Rosas–Sagan in the algebra NCSym to define quasisymmetric Schur functions in the algebra NCQSym. We prove these latter functions refine the former and their properties, and project onto quasisymmetric Schur functions under the forgetful map. Lastly, we show that by suitably labeling L C , skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets.

[1]  Adriano M. Garsia,et al.  A decomposition of Solomon's descent algebra , 1989 .

[2]  RICHARD P. STANLEY,et al.  On the Number of Reduced Decompositions of Elements of Coxeter Groups , 1984, Eur. J. Comb..

[3]  Jennifer Morse,et al.  A k-tableau characterization of k-Schur functions , 2005, math/0505519.

[4]  Daniel Krob,et al.  Noncommutative Symmetric Functions V: a degenerate Version of UQ(Gln) , 1999, Int. J. Algebra Comput..

[5]  Frank Sottile,et al.  Skew Littlewood–Richardson Rules from Hopf Algebras , 2009 .

[6]  Christophe Reutenauer,et al.  Algèbres de Hopf de tableaux , 1995 .

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

[8]  Grigori Olshanski,et al.  Shifted Schur Functions , 1996 .

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

[10]  Marcel Paul Schützenberger,et al.  La correspondance de Robinson , 1977 .

[11]  Frank Sottile,et al.  Skew Schubert functions and the Pieri formula for flag manifolds , 2001 .

[12]  HighWire Press.  Philosophical transactions of the Royal Society of London. Series A, Containing papers of a mathematical or physical character , 1896 .

[13]  Mike Zabrocki,et al.  Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables , 2008, Canadian Journal of Mathematics.

[14]  Peter McNamara.  Cylindric skew Schur functions , 2004 .

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

[16]  T. Tao,et al.  The honeycomb model of _{}(ℂ) tensor products I: Proof of the saturation conjecture , 1999 .

[17]  Frank Sottile,et al.  Combinatorial Hopf algebras and generalized Dehn–Sommerville relations , 2003, Compositio Mathematica.

[18]  Li Jin-q,et al.  Hopf algebras , 2019, Graduate Studies in Mathematics.

[19]  Daniel Krob,et al.  Noncommutative Symmetric Functions Iv: Quantum Linear Groups and Hecke Algebras at q = 0 , 1997 .

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

[21]  A. I. Molev,et al.  Littlewood-Richardson polynomials , 2007, 0704.0065.

[22]  Moritz Beckmann,et al.  Young tableaux , 2007 .

[23]  Victor Reiner,et al.  A quasisymmetric function for matroids , 2009, Eur. J. Comb..

[24]  Frank Sottile,et al.  Hopf Algebras and Edge-Labeled Posets , 1997 .

[25]  Louis J. Billera,et al.  Peak quasisymmetric functions and Eulerian enumeration , 2003, 0706.3486.

[26]  Sergey Fomin,et al.  Noncommutative schur functions and their applications , 2006, Discret. Math..

[27]  Stephanie van Willigenburg,et al.  Equality of Schur and Skew Schur Functions , 2004 .

[28]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[29]  Marcelo Aguiar,et al.  Coxeter Groups and Hopf Algebras , 2006 .

[30]  Sarah Mason,et al.  Refinements of the Littlewood-Richardson rule , 2009, 0908.3540.

[31]  Terence Tao,et al.  The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture , 1998, math/9807160.

[32]  Nathan Jacobson,et al.  Pi-algebras: an Introduction , 1975 .

[33]  Sarah Mason.  A DECOMPOSITION OF SCHUR FUNCTIONS AND AN ANALOGUE OF THE ROBINSON-SCHENSTED-KNUTH ALGORITHM , 2006 .

[34]  Louis Solomon,et al.  A Mackey formula in the group ring of a Coxeter group , 1976 .

[35]  Aaron Lauve,et al.  QSym over Sym has a stable basis , 2011, J. Comb. Theory, Ser. A.

[36]  Maria Chlouveraki,et al.  On Hecke Algebras , 2009 .

[37]  Sergey Fomin,et al.  A Littlewood-Richardson Miscellany , 1993, Eur. J. Comb..

[38]  L. Billera,et al.  Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions , 2004 .

[39]  John Milnor,et al.  On the Structure of Hopf Algebras , 1965 .

[40]  Bruce E. Sagan,et al.  Symmetric functions in noncommuting variables , 2002, math/0208168.

[41]  Frank Sottile,et al.  Noncommutative Pieri Operators on Posets , 2000, J. Comb. Theory, Ser. A.

[42]  Mike Zabrocki,et al.  THE HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND QUASI-SYMMETRIC FUNCTIONS IN NON-COMMUTATIVE VARIABLES ARE FREE AND CO-FREE , 2009 .

[43]  Christophe Tollu,et al.  Factorisation of Littlewood-Richardson coefficients , 2009, J. Comb. Theory, Ser. A.

[44]  Richard P. Stanley,et al.  Generalized Riffle Shuffles and Quasisymmetric Functions , 1999, math/9912025.

[45]  I. Schur,et al.  Ueber eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen , 1901 .

[46]  Louis J. Billera,et al.  Quasisymmetric functions and Kazhdan-Lusztig polynomials , 2007, 0710.3965.

[47]  Marcelo Aguiar,et al.  Monoidal Functors, Species, and Hopf Algebras , 2010 .

[48]  T. Inui,et al.  The Symmetric Group , 1990 .

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

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

[51]  Mark D. Haiman,et al.  Dual equivalence with applications, including a conjecture of Proctor , 1992, Discret. Math..

[52]  Christian Gutschwager.  Equality of multiplicity free skew characters , 2008 .

[53]  D. E. Littlewood,et al.  Group Characters and Algebra , 1934 .

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

[55]  R. Ehrenborg.  On Posets and Hopf Algebras , 1996 .

[56]  Susan Montgomery,et al.  Hopf algebras and their actions on rings , 1993 .

[57]  Jason Fulman.  Descent algebras, hyperplane arrangements, and shuffling cards , 1998 .

[58]  A. Lascoux.  Puissances extérieures, déterminants et cycles de Schubert , 1974 .

[59]  P. Diaconis,et al.  Trailing the Dovetail Shuffle to its Lair , 1992 .

[60]  Skew Schur Functions and Yangian Actions on Irreducible Integrable Modules of $ \hat{\frak g \frak l} _N $ , 2000 .

[61]  Victor Reiner,et al.  Coincidences among skew Schur functions , 2006 .

[62]  Andrei Zelevinsky,et al.  Triple Multiplicities for sl(r + 1) and the Spectrum of the Exterior Algebra of the Adjoint Representation , 1992 .

[63]  Glânffrwd P Thomas.  On Schensted's construction and the multiplication of schur functions , 1978 .

[64]  Skew Young Diagram Method in Spectral Decomposition of Integrable Lattice Models , 1996, q-alg/9607027.

[65]  Gérard Duchamp,et al.  Noncommutative Symmetric Functions Vi: Free Quasi-Symmetric Functions and Related Algebras , 2002, Int. J. Algebra Comput..

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

[67]  Stephanie van Willigenburg,et al.  Towards a Combinatorial Classification of Skew Schur Functions , 2006 .

[68]  Bruce E. Sagan,et al.  A Littlewood-Richardson rule for factorial Schur functions , 1997 .

[69]  Manfred Schocker.  Lie idempotent algebras , 2003 .

[70]  Sami H. Assaf.  Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity , 2010, 1005.3759.

[71]  S. Raianu,et al.  Hopf algebras : an introduction , 2001 .

[72]  Bruce E. Sagan,et al.  The symmetric group - representations, combinatorial algorithms, and symmetric functions , 2001, Wadsworth & Brooks / Cole mathematics series.

[73]  C. Reutenauer,et al.  Duality between Quasi-Symmetrical Functions and the Solomon Descent Algebra , 1995 .

[74]  D. Foata,et al.  Combinatoire et Représentation du Groupe Symétrique , 1977 .

[75]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.