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A Littlewood-type identity for Robbins polynomials
Fischer, Ilse1; Höngesberg, Hans2
1Fakultät für Mathematik, Universität Wien, 1090 Wien, Austria
2Fakulteta za matematiko in fiziko, Univerza v Ljubljani, 1000 Ljubljana, Slovenia (former)
Algebraic Combinatorics, Volume 9 (2026) no. 3, pp. 665-700

We provide a generalization of the Littlewood identity, both sides of which are related to alternating sign matrices. The classical Littlewood identity establishes a nice product formula for the sum of all Schur polynomials. Compared to the classical identity, Schur polynomials are replaced by so-called modified Robbins polynomials. These polynomials are a generalization of Schur polynomials and enumerate down-arrowed monotone triangles, and thus also alternating sign matrices. As an additional factor on the other side of the identity, we have a Pfaffian formula which we interpret in terms of the partition function of six-vertex model configurations corresponding to diagonally symmetric alternating sign matrices.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.495
Classification: 05A05, 05A15, 05A19, 05E05, 82B20
Keywords: Littlewood identity, alternating sign matrices, monotone triangles, six-vertex model

Fischer, Ilse  1 ; Höngesberg, Hans  2

1 Fakultät für Mathematik, Universität Wien, 1090 Wien, Austria
2 Fakulteta za matematiko in fiziko, Univerza v Ljubljani, 1000 Ljubljana, Slovenia (former)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Fischer, Ilse; Höngesberg, Hans. A Littlewood-type identity for Robbins polynomials. Algebraic Combinatorics, Volume 9 (2026) no. 3, pp. 665-700. doi: 10.5802/alco.495
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[1] Andrews, George E. Plane partitions. III. The weak Macdonald conjecture, Invent. Math., Volume 53 (1979) no. 3, pp. 193-225 | Zbl | DOI | MR

[2] Andrews, George E. Plane partitions. V. The TSSCPP conjecture, J. Combin. Theory Ser. A, Volume 66 (1994) no. 1, pp. 28-39 | Zbl | DOI | MR

[3] Ayyer, Arvind; Behrend, Roger E.; Fischer, Ilse Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order, Adv. Math., Volume 367 (2020), Paper no. 107125, 56 pages | Zbl | DOI | MR

[4] Behrend, Roger E.; Fischer, Ilse; Koutschan, Christoph Diagonally symmetric alternating sign matrices, 2023 | arXiv | Zbl

[5] Betea, D.; Wheeler, M. Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices, J. Combin. Theory Ser. A, Volume 137 (2016), pp. 126-165 | Zbl | DOI | MR

[6] Betea, D.; Wheeler, M.; Zinn-Justin, P. Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures, J. Algebraic Combin., Volume 42 (2015) no. 2, pp. 555-603 | Zbl | DOI | MR

[7] Borodin, Alexei; Petrov, Leonid Higher spin six vertex model and symmetric rational functions, Selecta Math. (N.S.), Volume 24 (2018) no. 2, pp. 751-874 | Zbl | DOI | MR

[8] Bressoud, David; Propp, James How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., Volume 46 (1999) no. 6, pp. 637-646 | Zbl | MR

[9] Bressoud, David M. Elementary proofs of identities for Schur functions and plane partitions, Ramanujan J., Volume 4 (2000) no. 1, pp. 69-80 | Zbl | DOI | MR

[10] Ciucu, Mihai Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A, Volume 77 (1997) no. 1, pp. 67-97 | Zbl | DOI | MR

[11] Elkies, Noam; Kuperberg, Greg; Larsen, Michael; Propp, James Alternating-sign matrices and domino tilings. I, J. Algebraic Combin., Volume 1 (1992) no. 2, pp. 111-132 | Zbl | DOI | MR

[12] Elkies, Noam; Kuperberg, Greg; Larsen, Michael; Propp, James Alternating-sign matrices and domino tilings. II, J. Algebraic Combin., Volume 1 (1992) no. 3, pp. 219-234 | Zbl | DOI | MR

[13] Fischer, Ilse A new proof of the refined alternating sign matrix theorem, J. Combin. Theory Ser. A, Volume 114 (2007) no. 2, pp. 253-264 | Zbl | DOI | MR

[14] Fischer, Ilse A constant term approach to enumerating alternating sign trapezoids, Adv. Math., Volume 356 (2019), Paper no. 106792, 23 pages | Zbl | DOI | MR

[15] Fischer, Ilse Enumeration of alternating sign triangles using a constant term approach, Trans. Amer. Math. Soc., Volume 372 (2019) no. 2, pp. 1485-1508 | Zbl | DOI | MR

[16] Fischer, Ilse Computations versus bijections for tiling enumeration, Adv. in Appl. Math., Volume 142 (2023), Paper no. 102427, 49 pages | DOI | MR | Zbl

[17] Fischer, Ilse Bounded Littlewood identity related to alternating sign matrices, Forum Math. Sigma, Volume 12 (2024), Paper no. e124, 44 pages | DOI | MR | Zbl

[18] Fischer, Ilse; Konvalinka, Matjaž The mysterious story of square ice, piles of cubes, and bijections, Proc. Natl. Acad. Sci. USA, Volume 117 (2020) no. 38, pp. 23460-23466 | Zbl | DOI | MR

[19] Fischer, Ilse; Riegler, Lukas Combinatorial reciprocity for monotone triangles, J. Combin. Theory Ser. A, Volume 120 (2013) no. 7, pp. 1372-1393 | Zbl | DOI | MR

[20] Fischer, Ilse; Schreier-Aigner, Florian The relation between alternating sign matrices and descending plane partitions: n+3 pairs of equivalent statistics, Adv. Math., Volume 413 (2023), p. Paper No. 108831, 47 | Zbl | DOI | MR

[21] Gangl, Moritz Alternating sign pentagons and Magog pentagons, Adv. Math., Volume 474 (2025), Paper no. 110315, 29 pages | Zbl | DOI | MR

[22] Gavrilova, Svetlana Refined Littlewood identity for spin Hall-Littlewood symmetric rational functions, Algebr. Comb., Volume 6 (2023) no. 1, pp. 37-51 | Numdam | Zbl | DOI | MR

[23] Höngesberg, Hans Refined enumeration of halved monotone triangles and applications to vertically symmetric alternating sign trapezoids, J. Combin. Theory Ser. A, Volume 177 (2021), Paper no. 105336, 37 pages | Zbl | DOI | MR

[24] Höngesberg, Hans A fourfold refined enumeration of alternating sign trapezoids, Electron. J. Combin., Volume 29 (2022) no. 3, Paper no. 3.42, 27 pages | Zbl | DOI | MR

[25] Ikeda, Takeshi; Naruse, Hiroshi K-theoretic analogues of factorial Schur P- and Q-functions, Adv. Math., Volume 243 (2013), pp. 22-66 | Zbl | DOI | MR

[26] Kawanaka, Noriaki On subfield symmetric spaces over a finite field, Osaka J. Math., Volume 28 (1991) no. 4, pp. 759-791 | Zbl | MR

[27] Kawanaka, Noriaki A q-series identity involving Schur functions and related topics, Osaka J. Math., Volume 36 (1999) no. 1, pp. 157-176 | Zbl | MR

[28] Krattenthaler, C. Plane partitions in the work of Richard Stanley and his school, The mathematical legacy of Richard P. Stanley, Amer. Math. Soc., Providence, RI, 2016, pp. 231-261 | Zbl | DOI | MR

[29] Krattenthaler, Christian A Gog-Magog conjecture, 1996 https://www.mat.univie.ac.at/... (unpublished note)

[30] Kuperberg, Greg Another proof of the alternating-sign matrix conjecture, Internat. Math. Res. Notices (1996) no. 3, pp. 139-150 | Zbl | DOI | MR

[31] Littlewood, Dudley E. The Theory of Group Characters and Matrix Representations of Groups, Oxford University Press, New York, 1940, viii+292 pages | Zbl | MR

[32] Mills, W. H.; Robbins, David P.; Rumsey, Howard Jr. Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A, Volume 34 (1983) no. 3, pp. 340-359 | Zbl | DOI | MR

[33] Mills, W. H.; Robbins, David P.; Rumsey, Howard Jr. Self-complementary totally symmetric plane partitions, J. Combin. Theory Ser. A, Volume 42 (1986) no. 2, pp. 277-292 | Zbl | DOI | MR

[34] Petrov, Leonid Refined Cauchy identity for spin Hall-Littlewood symmetric rational functions, J. Combin. Theory Ser. A, Volume 184 (2021), Paper no. 105519, 50 pages | Zbl | DOI | MR

[35] Propp, James Generalized domino-shuffling, Theoret. Comput. Sci., Volume 303 (2003) no. 2-3, pp. 267-301 | Zbl | DOI | MR

[36] Robbins, David P. The story of 1,2,7,42,429,7436,⋯, Math. Intelligencer, Volume 13 (1991) no. 2, pp. 12-19 | MR | Zbl | DOI

[37] Robbins, David P. Symmetry classes of alternating sign matrices, 2000 | arXiv | Zbl

[38] Robbins, David P.; Rumsey, Howard Jr. Determinants and alternating sign matrices, Adv. in Math., Volume 62 (1986) no. 2, pp. 169-184 | DOI | MR | Zbl

[39] Schur, J. Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., Volume 139 (1911), pp. 155-250 | DOI | MR | Zbl

[40] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | Zbl | MR

[41] Warnaar, S. Ole Bisymmetric functions, Macdonald polynomials and 𝔰𝔩 3 basic hypergeometric series, Compos. Math., Volume 144 (2008) no. 2, pp. 271-303 | Zbl | DOI | MR

[42] Wheeler, Michael; Zinn-Justin, Paul Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons, Adv. Math., Volume 299 (2016), pp. 543-600 | Zbl | DOI | MR

[43] Zeilberger, Doron Proof of the alternating sign matrix conjecture, Electron. J. Combin., Volume 3 (1996) no. 2, Paper no. 13, 84 pages | Zbl | DOI | MR

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