Abstract
We define an action of the 0-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their (q, t)-analogues introduced by Bergeron and Zabrocki, and to a more general family of noncommutative symmetric functions having parameters associated with paths in binary trees introduced recently by Lascoux, Novelli, and Thibon. We also obtain multivariate quasisymmetric function identities, which specialize to results of Garsia and Gessel on generating functions of multivariate distributions of permutation statistics.
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Adin R.M., Brenti F., Roichman Y.: Descent representations and multivariate statistics. Trans. Amer. Math. Soc. 357(8), 3051–3082 (2005)
Adin R.M., Postnikov A., Roichman Y.: Hecke algebra actions on the coinvariant algebra. J. Algebra 233(2), 594–613 (2000)
Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1: Techniques of Representation Theory. London Math. Soc. Stud. Texts, Vol. 65. Cambridge University Press, Cambridge (2006)
Bergeron N., Li H.: Algebraic structures on Grothendieck groups of a tower of algebras. J. Algebra 321(8), 2068–2084 (2009)
Bergeron N., Zabrocki M.: q and q, t-analogs of non-commutative symmetric functions. Discrete Math. 298(1-3), 79–103 (2005)
Björner A.: Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings. Adv. Math. 52(3), 173–212 (1984)
Björner A., Wachs M.L.: Generalized quotients in Coxeter groups. Trans. Amer. Math. Soc. 308(1), 1–37 (1988)
Carlitz L.: A combinatorial property of q-Eulerian numbers. Amer. Math. Monthly 82(1), 51–54 (1975)
Garsia A.M.: Combinatorial methods in the theory of Cohen-Macaulay rings. Adv. Math. 38(3), 229–266 (1980)
Garsia A.M., Gessel I.: Permutation statistics and partitions. Adv. Math. 31(3), 288–305 (1979)
Garsia A.M., Procesi C.: On certain graded Sn-modules and the q-Kostka polynomials. Adv. Math. 94(1), 82–138 (1992)
Garsia A.M., Stanton D.: Group actions of Stanley-Reisner rings and invariant of permutation groups. Adv. Math. 51(2), 107–201 (1984)
Gessel I.M., Reutenauer C.: Counting permutations with given cycle structure and descent set. J. Combin. Theory Ser. A 64(2), 189–215 (1993)
Hivert, F., Lascoux, A., Thibon, J.-Y.: Noncommutative symmetric functions and quasisymmetric functions with two and more parameters. Preprint, arXiv:math/0106191 (2001)
Hivert, F., Thiéry, N.M.: The Hecke group algebra of a Coxeter group and its representation theory. J. Algebra 321(8), 2230–2258 (2009)
Hotta R., Springer T.A.: A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups. Invent. Math. 41(2), 113–127 (1977)
Huang J.: 0-Hecke algebra actions on coinvariants and flags. J. Algebraic Combin. 40(1), 245–278 (2014)
Huang, J.: 0-Hecke algebra actions on flags, polynomials, and Stanley-Reisner rings. Doctoral Dissertation. University of Minnesota, Minneapolis (2013)
Kind B., Kleinschmidt P.: Schälbare Cohen-Macauley-Komplexe und ihre Parametrisierung. Math. Z. 167(2), 173–179 (1979)
Krob, D., Thibon, J.-Y.: Noncommutative symmetric functions IV: quantum linear groups and Hecke algebras at q = 0. J. Algebraic Combin. 6(4), 339–376 (1997)
Lascoux, A., Novelli, J.-C., Thibon, J.-Y.: Noncommutative symmetric functions with matrix parameters. J. Algebraic Combin. 37(4), 621–642 (2013)
Lauda, A.D., Russell, H.M.: Oddification of the cohomology of type A Springer varieties. Int. Math. Res. Not. IMRN 2014(17), 4822–4854 (2014)
MacMahon, P.A.: Combinatorial Analysis, Vols. 1 and 2. Cambridge University Press, Cambridge (1915–1916)
Mathas A.: A q-analogue of the Coxeter complex. J. Algebra 164(3), 831–848 (1994)
Norton P.N.: 0-Hecke algebras. J. Austral. Math. Soc. Ser. A 27, 337–357 (1979)
Stanley, R.P.: Enumerative Combinatorics. Vol. 2. Cambridge University Press, Cambridge (1999)
Tanisaki T.: Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups. Tôhoku Math. J. 34, 575–585 (1982)
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Huang, J. 0-Hecke Algebra Action on the Stanley-Reisner Ring of the Boolean Algebra. Ann. Comb. 19, 293–323 (2015). https://doi.org/10.1007/s00026-015-0264-y
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DOI: https://doi.org/10.1007/s00026-015-0264-y
Keywords
- 0-Hecke algebra
- Stanley-Reisner ring
- Boolean algebra
- noncommutative Hall-Littlewood symmetric function
- multivariate quasisymmetric function