Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Pieri operators on the affine nilCoxeter algebra


Authors: Chris Berg, Franco Saliola and Luis Serrano
Journal: Trans. Amer. Math. Soc. 366 (2014), 531-546
MSC (2010): Primary 05E05; Secondary 14N15
DOI: https://doi.org/10.1090/S0002-9947-2013-05895-3
Published electronically: June 18, 2013
MathSciNet review: 3118405
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study a family of operators on the affine nilCoxeter algebra. We use these operators to prove conjectures of Lam, Lapointe, Morse, and Shimozono regarding strong Schur functions.


References [Enhancements On Off] (What's this?)

  • [BSS12] Chris Berg, Franco Saliola, and Luis Serrano, The down operator and expansions of near rectangular $ k$-Schur functions, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), Discrete Math. Theor. Comput. Sci. Proc., AR, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2012, pp. 433-444 (English, with English and French summaries). MR 2958018
  • [BMSvW00] Nantel Bergeron, Stefan Mykytiuk, Frank Sottile, and Stephanie van Willigenburg, Noncommutative Pieri operators on posets, J. Combin. Theory Ser. A 91 (2000), no. 1-2, 84-110. In memory of Gian-Carlo Rota. MR 1779776 (2002d:05122), https://doi.org/10.1006/jcta.2000.3090
  • [BTvW06] Louis J. Billera, Hugh Thomas, and Stephanie van Willigenburg, Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions, Adv. Math. 204 (2006), no. 1, 204-240. MR 2233132 (2007b:05207), https://doi.org/10.1016/j.aim.2005.05.014
  • [Lam10] T. Lam, Stanley symmetric functions and Peterson algebras (2010). ArXiv e-prints.
  • [Lam06] Thomas Lam, Affine Stanley symmetric functions, Amer. J. Math. 128 (2006), no. 6, 1553-1586. MR 2275911 (2008b:05178)
  • [Lam08] Thomas Lam, Schubert polynomials for the affine Grassmannian, J. Amer. Math. Soc. 21 (2008), no. 1, 259-281. MR 2350056 (2009a:05207), https://doi.org/10.1090/S0894-0347-06-00553-4
  • [LLMS10] Thomas Lam, Luc Lapointe, Jennifer Morse, and Mark Shimozono, Affine insertion and Pieri rules for the affine Grassmannian, Mem. Amer. Math. Soc. 208 (2010), no. 977, xii+82. MR 2741963 (2012f:05314), https://doi.org/10.1090/S0065-9266-10-00576-4
  • [LS10] Thomas Lam and Mark Shimozono, Quantum cohomology of $ G/P$ and homology of affine Grassmannian, Acta Math. 204 (2010), no. 1, 49-90. MR 2600433 (2011h:14082), https://doi.org/10.1007/s11511-010-0045-8
  • [LS07] Thomas F. Lam and Mark Shimozono, Dual graded graphs for Kac-Moody algebras, Algebra Number Theory 1 (2007), no. 4, 451-488. MR 2368957 (2009a:05209), https://doi.org/10.2140/ant.2007.1.451
  • [LLM03] L. Lapointe, A. Lascoux, and J. Morse, Tableau atoms and a new Macdonald positivity conjecture, Duke Math. J. 116 (2003), no. 1, 103-146. MR 1950481 (2004c:05208), https://doi.org/10.1215/S0012-7094-03-11614-2
  • [LM03] L. Lapointe and J. Morse, Schur function analogs for a filtration of the symmetric function space, J. Combin. Theory Ser. A 101 (2003), no. 2, 191-224. MR 1961543 (2004c:05209), https://doi.org/10.1016/S0097-3165(02)00012-2
  • [LM05] Luc Lapointe and Jennifer Morse, Tableaux on $ k+1$-cores, reduced words for affine permutations, and $ k$-Schur expansions, J. Combin. Theory Ser. A 112 (2005), no. 1, 44-81. MR 2167475 (2006j:05214), https://doi.org/10.1016/j.jcta.2005.01.003
  • [LM07] Luc Lapointe and Jennifer Morse, A $ k$-tableau characterization of $ k$-Schur functions, Adv. Math. 213 (2007), no. 1, 183-204. MR 2331242 (2008c:05187), https://doi.org/10.1016/j.aim.2006.12.005
  • [LM08] Luc Lapointe and Jennifer Morse, Quantum cohomology and the $ k$-Schur basis, Trans. Amer. Math. Soc. 360 (2008), no. 4, 2021-2040. MR 2366973 (2009d:14072), https://doi.org/10.1090/S0002-9947-07-04287-0
  • [LL12] Naichung Conan Leung and Changzheng Li, Gromov-Witten invariants for $ G/B$ and Pontryagin product for $ \Omega K$, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2567-2599. MR 2888220 (2012m:14107), https://doi.org/10.1090/S0002-9947-2012-05438-9
  • [Mac] P.A. MacMahon, Combinatory analysis, Combinatory Analysis, no. v. 2. The University Press, 1916.
  • [SCc12] The Sage-Combinat community, Sage-Combinat: enhancing sage as a toolbox for computer exploration in algebraic combinatorics, The Sage Development Team (2012).
  • [Sta] 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 (2000k:05026)
  • [Ste82] E. M. Stein, The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 359-376. MR 663787 (83i:42001), https://doi.org/10.1090/S0273-0979-1982-15040-6

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 05E05, 14N15

Retrieve articles in all journals with MSC (2010): 05E05, 14N15


Additional Information

Chris Berg
Affiliation: Department of Mathematics, University of Quebec at Montreal, Montreal, Quebec, Canada H8C 3P8

Franco Saliola
Affiliation: Department of Mathematics, University of Quebec at Montreal, Montreal, Quebec, Canada H8C 3P8

Luis Serrano
Affiliation: Department of Mathematics, University of Quebec at Montreal, Montreal, Quebec, Canada H8C 3P8

DOI: https://doi.org/10.1090/S0002-9947-2013-05895-3
Received by editor(s): March 29, 2012
Received by editor(s) in revised form: June 12, 2012
Published electronically: June 18, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society