Skip to main content
SpringerLink
Log in

Alternating Sign Matrices: Extensions, König-Properties, and Primary Sum-Sequences

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

This paper is concerned with properties of permutation matrices and alternating sign matrices (ASMs). An ASM is a square \((0,\pm 1)\)-matrix such that, ignoring 0’s, the 1’s and \(-1\)’s in each row and column alternate, beginning and ending with a 1. We study extensions of permutation matrices into ASMs by changing some zeros to \(+1\) or \(-1\). Furthermore, several properties concerning the term rank and line covering of ASMs are shown. An ASM A is determined by a sum-matrix \(\varSigma (A)\) whose entries are the sums of the entries of its leading submatrices (so determined by the entries of A). We show that those sums corresponding to the nonzero entries of a permutation matrix determine all the entries of the sum-matrix and investigate some of the properties of the resulting sequence of numbers. Finally, we investigate the lattice-properties of the set of ASMs (of order n), where the partial order comes from the Bruhat order for permutation matrices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)

    MATH  Google Scholar 

  2. Bóna, M.: Combinatorics of Permutations, 2nd edn. CRC Press, Boca Raton (2012)

    MATH  Google Scholar 

  3. Brualdi, R.A., Dahl, G.: The Bruhat shadow of a permutation matrix, Mathematical Papers in Honour of Eduardo Marques de Sá, O, Azenhas, Duarte, A.L., Queiró, J.F., Santana, A.P., eds., Textos de Mathemática Vol. 39, Dep. de Matemática da Univ. de Coimbra, Portugal, pp. 25–38 (2006)

  4. Brualdi, R.A., Dahl, G.: Alternating sign matrices, extensions and related cones. Adv. Appl. Math. 86, 19–49 (2017)

    Article  MathSciNet  Google Scholar 

  5. Brualdi, R.A., Dahl, G.: Alternating sign matrices and hypermatrices, and a generalization of latin squares. Adv. Appl. Math. 95, 116–151 (2018)

    Article  MathSciNet  Google Scholar 

  6. Brualdi, R.A., Kim, H.K.: Completions of alternating sign matrices. Graphs Combin. 31, 507–522 (2015)

    Article  MathSciNet  Google Scholar 

  7. Brualdi, R.A., Kiernan, K.P., Meyer, S.A., Schroeder, M.W.: Patterns of alternating sign matrices. Linear Algebra Appl. 438, 3967–3990 (2013)

    Article  MathSciNet  Google Scholar 

  8. Brualdi, R.A., Schroeder, M.W.: Alternating sign matrices and their Bruhat order. Discr. Math. 340, 1996–2019 (2017)

    Article  MathSciNet  Google Scholar 

  9. Eriksson, K., Linusson, S.: The size of Fulton’s essential set. Electr. J. Combin. 2, R6 (1995)

  10. Eriksson, K., Linusson, S.: Combinatorics of Fulton’s essential set. Duke Math. J. 85, 61–76 (1996)

    Article  MathSciNet  Google Scholar 

  11. Fulton, W.: Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. Duke Math. J. 65, 381–420 (1992)

    Article  MathSciNet  Google Scholar 

  12. Lascoux, A., Schützenberger, M.P.: Treillis et bases des groupes de Coxeter. Electron. J. Combin. 3, Research paper 27 (1996)

  13. Terwilliger, P.: A poset \(\Phi _n\) whose maximal chains are in bijection with the \(n\times n\) alternating sign matrices. Linear Algebra Appl. 554, 79–85 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank a referee for several useful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Richard A. Brualdi

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Geir Dahl

Authors
  1. Richard A. Brualdi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Geir Dahl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Geir Dahl.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brualdi, R.A., Dahl, G. Alternating Sign Matrices: Extensions, König-Properties, and Primary Sum-Sequences. Graphs and Combinatorics 36, 63–92 (2020). https://doi.org/10.1007/s00373-019-02119-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-019-02119-x

Keywords

Search

Navigation