Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED ALTERNATING SIGN MATRICES AND SIGNED PERMUTATION MATRICES

  • Brualdi, Richard A. (Department of Mathematics University of Wisconsin) ;
  • Kim, Hwa Kyung (Department of Mathematics Education Sangmyung University)
  • Received : 2020.06.04
  • Accepted : 2020.08.31
  • Published : 2021.07.01
Download PDF
⟨ Previous Next ⟩

Abstract

We continue the investigations in [6] extending the Bruhat order on n × n alternating sign matrices to our more general setting. We show that the resulting partially ordered set is a graded lattice with a well-define rank function. Many illustrative examples are given.

Keywords

Acknowledgement

We are very grateful to Roger Behrend who provided us with extensive and valuable comments in the preparation of this paper.

References

  1. R. E. Behrend and V. A. Knight, Higher spin alternating sign matrices, Electron. J. Combin. 14 (2007), no. 1, Research Paper 83, 38 pp.
  2. A. Bjorner and F. Brenti. Combinatorics of Coxeter Groups, Graduate Texts in Math. Vol. 231, Springer, 2003.
  3. D. M. Bressoud, Proofs and Confirmations, MAA Spectrum, Mathematical Association of America, Washington, DC, 1999.
  4. R. A. Brualdi and G. Dahl, Alternating sign matrices, extensions and related cones, Adv. in Appl. Math. 86 (2017), 19-49. https://doi.org/10.1016/j.aam.2016.12.001
  5. R. A. Brualdi, K. P. Kiernan, S. A. Meyer, and M. W. Schroeder, Patterns of alternating sign matrices, Linear Algebra Appl. 438 (2013), no. 10, 3967-3990. https://doi.org/10.1016/j.laa.2012.03.009
  6. R. A. Brualdi and H. K. Kim, A generalization of alternating sign matrices, J. Combin. Des. 23 (2015), no. 5, 204-215. https://doi.org/10.1002/jcd.21397
  7. R. A. Brualdi and M. W. Schroeder, Alternating sign matrices and their Bruhat order, Discrete Math. 340 (2017), no. 8, 1996-2019. https://doi.org/10.1016/j.disc.2016.10.010
  8. Z. Hamaker and V. Reiner, Weak order and descents for monotone triangles, European J. Combin. 86 (2020), 103083, 22 pp. https://doi.org/10.1016/j.ejc.2020.103083
  9. A. Lascoux and M.-P. Schutzenberger, Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3 (1996), no. 2, Research paper 27, approx. 35 pp.
  10. W. H. Mills, D. P. Robbins, and H. Rumsey, Jr., Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), no. 3, 340-359. https://doi.org/10.1016/0097-3165(83)90068-7
  11. J. Propp, The many faces of alternating-sign matrices, in Discrete models: combinatorics, computation, and geometry (Paris, 2001), 043-058, Discrete Math. Theor. Comput. Sci. Proc., AA, Maison Inform. Math. Discr'et. (MIMD), Paris, 2001.