More Web Proxy on the site http://driver.im/

article

Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures

Authors:

P. Zinn-JustinAuthors Info & Claims

Journal of Algebraic Combinatorics: An International Journal, Volume 42, Issue 2

Pages 555 - 603

https://doi.org/10.1007/s10801-015-0592-3

Published: 01 September 2015 Publication History

Abstract

We prove two identities of Hall---Littlewood polynomials, which appeared recently in Betea and Wheeler (2014). We also conjecture, and in some cases prove, new identities which relate infinite sums of symmetric polynomials and partition functions associated with symmetry classes of alternating sign matrices. These identities generalize those already found in Betea and Wheeler (2014), via the introduction of additional parameters. The left-hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity. The right-hand side of each identity is (one of the two factors present in) the partition function of the six-vertex model on a relevant domain.

References

[1]

Andrews, G.E., Goulden, I.P., Jackson, D.M.: Generalizations of Cauchy's summation theorem for Schur functions. Trans. Am. Math. Soc. 310(2), 805---820 (1988)

[2]

Betea, D., Wheeler, M.: Refined Cauchy and littlewood identities, plane partitions, and symmetry classes of alternating sign matrices (2014). arXiv:1402.0229

[3]

Izergin, A.G.: Partition function of the six-vertex model in a finite volume. Dokl. Akad. Nauk SSSR 297(2), 331---333 (1987)

[4]

Kirillov, A.N., Noumi, M.: $$q$$q-difference raising operators for Macdonald polynomials and the integrality of transition coefficients. In: Algebraic Methods and $$q$$q-Special Functions (Montréal, QC, 1996), CRM Proc. Lecture Notes, vol. 22. Am. Math. Soc, Providence, RI, pp. 227---243 (1999)

[5]

Korepin, V., Zinn-Justin, P.: Thermodynamic limit of the six-vertex model with domain wall boundary conditions. J. Phys. A 33(40), 7053---7066 (2000)

[6]

Korepin, V.E.: Calculation of norms of Bethe wave functions. Commun. Math. Phys. 86(3), 391---418 (1982)

[7]

Kuperberg, G.: Another proof of the alternating-sign matrix conjecture. Int. Math. Res. Not. 3, 139---150 (1996)

[8]

Kuperberg, G.: Symmetry classes of alternating-sign matrices under one roof. Ann. Math. 156(3), 835---866 (2002)

[9]

Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. The Clarendon Press, Oxford University Press, New York (1995). With contributions by A Zelevinsky, Oxford Science Publications (1995)

[10]

Okounkov, A.: BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials. Transform. Groups 3(2), 181---207 (1998)

[11]

Rains, E.M.: $$\text{ BC }_n$$BCn-symmetric polynomials. Transform. Groups 10(1), 63---132 (2005)

[12]

Sklyanin, E.K.: Boundary conditions for integrable quantum systems. J. Phys. A 21(10), 2375---2389 (1988)

[13]

Sundaram, S.: Tableaux in the representation theory of the classical Lie groups. In: Invariant Theory and Tableaux (Minneapolis, MN, 1988), IMA Vol. Math. Appl., vol. 19. Springer, New York, pp. 191---225 (1990)

[14]

Tsuchiya, O.: Determinant formula for the six-vertex model with reflecting end. J. Math. Phys. 39(11), 5946---5951 (1998)

[15]

Venkateswaran, V.: Symmetric and nonsymmetric Hall---Littlewood polynomials of type BC (2013). arXiv:1209.2933v2

[16]

Vuletić, M.: A generalization of MacMahon's formula. Trans. Am. Math. Soc. 361(5), 2789---2804 (2009)

[17]

Warnaar, S.O.: Bisymmetric functions, Macdonald polynomials and $$sl_{3}$$sl3 basic hypergeometric series. Compos. Math. 144(2), 271---303 (2008)

[18]

Warnaar, S.O.: Remarks on the paper "Skew Pieri rules for Hall---Littlewood functions" by Konvalinka and Lauve. J. Algebr. Comb. 38(3), 519---526 (2013)

Digital Library

Cited By

Betea DWheeler M(2016)Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matricesJournal of Combinatorial Theory Series A10.1016/j.jcta.2015.08.007137:C(126-165)Online publication date: 1-Jan-2016
https://dl.acm.org/doi/10.1016/j.jcta.2015.08.007

Recommendations

Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices

We prove and conjecture some new symmetric function identities, which equate the generating series of 1. Plane partitions, subject to certain restrictions and weightings, and 2. Alternating sign matrices, subject to certain symmetry properties. The left ...
Elliptic Littlewood identities

We prove analogues for elliptic interpolation functions of Macdonald@?s version of the Littlewood identity for (skew) Macdonald polynomials, in the process developing an interpretation of general elliptic ''hypergeometric'' sums as skew interpolation ...
A generalization of Kawanaka's identity for Hall-Littlewood polynomials and applications

An infinite summation formula of Hall-Littlewood polynomials due to Kawanaka is generalized to a finite summation formula, which implies, in particular, twelve more multiple q -identities of Rogers-Ramanujan type than those previously found by ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Journal of Algebraic Combinatorics: An International Journal

Journal of Algebraic Combinatorics: An International Journal Volume 42, Issue 2

September 2015

336 pages

ISSN:0925-9899

Issue’s Table of Contents

Copyright © Copyright © 2015 Springer Science+Business Media New York.

Publisher

Kluwer Academic Publishers

United States

Publication History

Published: 01 September 2015

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

1
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 19 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Betea DWheeler M(2016)Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matricesJournal of Combinatorial Theory Series A10.1016/j.jcta.2015.08.007137:C(126-165)Online publication date: 1-Jan-2016
https://dl.acm.org/doi/10.1016/j.jcta.2015.08.007

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents