Descent representations and multivariate statistics

Adin, Ron; Brenti, Francesco; Roichman, Yuval

doi:10.1090/S0002-9947-04-03494-4

Descent representations and multivariate statistics
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by Ron M. Adin, Francesco Brenti and Yuval Roichman PDF

Trans. Amer. Math. Soc. 357 (2005), 3051-3082 Request permission

Abstract:

Combinatorial identities on Weyl groups of types $A$ and $B$ are derived from special bases of the corresponding coinvariant algebras. Using the Garsia-Stanton descent basis of the coinvariant algebra of type $A$ we give a new construction of the Solomon descent representations. An extension of the descent basis to type $B$, using new multivariate statistics on the group, yields a refinement of the descent representations. These constructions are then applied to refine well-known decomposition rules of the coinvariant algebra and to generalize various identities.

References

R. M. Adin, F. Brenti and Y. Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. Appl. Math. 27 (2001), 210–224.
R. M. Adin and Y. Roichman, A flag major index for signed permutations, Proc. 11-th Conference on Formal Power Series and Algebraic Combinatorics, Universitat Politècnica de Catalunya, Barcelona, 1999, 10–17.
Ron M. Adin and Yuval Roichman, The flag major index and group actions on polynomial rings, European J. Combin. 22 (2001), no. 4, 431–446. MR 1829737, DOI 10.1006/eujc.2000.0469
E. E. Allen, The descent monomials and a basis for the diagonally symmetric polynomials, J. Algebraic Combin. 3 (1994), no. 1, 5–16. MR 1256100, DOI 10.1023/A:1022481303750
Susumu Ariki, Tomohide Terasoma, and Hiro-Fumi Yamada, Higher Specht polynomials, Hiroshima Math. J. 27 (1997), no. 1, 177–188. MR 1437932
Hélène Barcelo, Young straightening in a quotient $S_n$-module, J. Algebraic Combin. 2 (1993), no. 1, 5–23. MR 1210398, DOI 10.1023/A:1022416129423
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Schubert cells, and the cohomology of the spaces $G/P$, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3–26 (Russian). MR 0429933
A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, Springer-Verlag, to appear.
Francesco Brenti, $q$-Eulerian polynomials arising from Coxeter groups, European J. Combin. 15 (1994), no. 5, 417–441. MR 1292954, DOI 10.1006/eujc.1994.1046
Michel Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301 (French). MR 342522, DOI 10.1007/BF01418790
D. Foata, personal communication, July 2000.
Dominique Foata and Guo-Niu Han, Calcul basique des permutations signées. I. Longueur et nombre d’inversions, Adv. in Appl. Math. 18 (1997), no. 4, 489–509 (French, with English and French summaries). MR 1445359, DOI 10.1006/aama.1997.0525
Adriano M. Garsia, Combinatorial methods in the theory of Cohen-Macaulay rings, Adv. in Math. 38 (1980), no. 3, 229–266. MR 597728, DOI 10.1016/0001-8708(80)90006-7
Adriano M. Garsia, On the “maj” and “inv” $q$-analogues of Eulerian polynomials, Linear and Multilinear Algebra 8 (1979/80), no. 1, 21–34. MR 548671, DOI 10.1080/03081087908817296
A. M. Garsia and C. Procesi, On certain graded $S_n$-modules and the $q$-Kostka polynomials, Adv. Math. 94 (1992), no. 1, 82–138. MR 1168926, DOI 10.1016/0001-8708(92)90034-I
A. M. Garsia and J. Remmel, Shuffles of permutations and the Kronecker product, Graphs Combin. 1 (1985), no. 3, 217–263. MR 951014, DOI 10.1007/BF02582950
A. M. Garsia and D. Stanton, Group actions of Stanley-Reisner rings and invariants of permutation groups, Adv. in Math. 51 (1984), no. 2, 107–201. MR 736732, DOI 10.1016/0001-8708(84)90005-7
I. M. Gessel, Generating functions and enumeration of sequences, Ph.D. Thesis, M.I.T., 1977.
Ira M. Gessel, Multipartite $P$-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289–317. MR 777705, DOI 10.1090/conm/034/777705
Howard Hiller, Geometry of Coxeter groups, Research Notes in Mathematics, vol. 54, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 649068
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
Witold Kraśkiewicz and Jerzy Weyman, Algebra of coinvariants and the action of a Coxeter element, Bayreuth. Math. Schr. 63 (2001), 265–284. MR 1867283
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Percy A. MacMahon, Combinatory analysis, Chelsea Publishing Co., New York, 1960. Two volumes (bound as one). MR 0141605
H. Morita and H.-F. Yamada, Higher Specht polynomials for the complex reflection group $G(r,p,n)$, Hokkaido Math. J. 27 (1998), no. 3, 505–515. MR 1662953, DOI 10.14492/hokmj/1351001459
Victor Reiner, Signed permutation statistics, European J. Combin. 14 (1993), no. 6, 553–567. MR 1248063, DOI 10.1006/eujc.1993.1058
Christophe Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1231799
Bruce E. Sagan, The symmetric group, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. Representations, combinatorial algorithms, and symmetric functions. MR 1093239
Louis Solomon, The orders of the finite Chevalley groups, J. Algebra 3 (1966), 376–393. MR 199275, DOI 10.1016/0021-8693(66)90007-X
T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279–293. MR 491988, DOI 10.1007/BF01403165
Richard P. Stanley, Ordered structures and partitions, Memoirs of the American Mathematical Society, No. 119, American Mathematical Society, Providence, R.I., 1972. MR 0332509
Richard P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 3, 475–511. MR 526968, DOI 10.1090/S0273-0979-1979-14597-X
Richard P. Stanley, Some aspects of groups acting on finite posets, J. Combin. Theory Ser. A 32 (1982), no. 2, 132–161. MR 654618, DOI 10.1016/0097-3165(82)90017-6
Richard P. Stanley, Enumerative combinatorics. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986. With a foreword by Gian-Carlo Rota. MR 847717, DOI 10.1007/978-1-4615-9763-6
Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
Robert Steinberg, On a theorem of Pittie, Topology 14 (1975), 173–177. MR 372897, DOI 10.1016/0040-9383(75)90025-7
John R. Stembridge, On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math. 140 (1989), no. 2, 353–396. MR 1023791, DOI 10.2140/pjm.1989.140.353
Tomohide Terasoma and Hirofumi Yamada, Higher Specht polynomials for the symmetric group, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 2, 41–44. MR 1210951