Some combinatorial aspects of reduced words in finite Coxeter groups

Stembridge, John

doi:10.1090/S0002-9947-97-01805-9

Some combinatorial aspects of reduced words in finite Coxeter groups
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by John R. Stembridge PDF

Trans. Amer. Math. Soc. 349 (1997), 1285-1332 Request permission

Abstract:

We analyze the structure of reduced expressions in the Coxeter groups $A_{n}$, $B_{n}$ and $D_{n}$. Several special classes of elements are singled out for their connections with symmetric functions or the theory of $P$-partitions. Membership in these special classes is characterized in a variety of ways, including forbidden patterns, forbidden subwords, and by the form of canonically chosen reduced words.

References

S. Billey and M. Haiman, Schubert polynomials for the classical groups, J. Amer. Math. Soc. 8 (1995), 443–482.
Sara C. Billey, William Jockusch, and Richard P. Stanley, Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2 (1993), no. 4, 345–374. MR 1241505, DOI 10.1023/A:1022419800503
Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
P. H. Edelman, Lexicographically first reduced words, Discrete Math. 147 (1995), 95–106.
C. K. Fan, “Commutative Elements of a Weyl Group”, Ph. D. thesis, MIT, 1995.
S. V. Fomin and A. N. Kirillov, Combinatorial $B_{n}$-analogues of Schubert polynomials, Trans. Amer. Math. Soc. 348 (1996), 3591–3620.
S. V. Fomin and A. N. Kirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math. 153 (1996), 123–143.
Sergey Fomin and Richard P. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), no. 2, 196–207. MR 1265793, DOI 10.1006/aima.1994.1009
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
T. K. Lam, “$B$ and $D$ Analogues of Stable Schubert Polynomials and Related Insertion Algorithms”, Ph. D. thesis, MIT, 1995.
I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 553598
Richard P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin. 5 (1984), no. 4, 359–372. MR 782057, DOI 10.1016/S0195-6698(84)80039-6
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
J. R. Stembridge, Enriched $P$-partitions, Trans. Amer. Math. Soc. 349 (1997), 763–788.
J. R. Stembridge, On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 5 (1996), 353-385.