Principal subspaces of basic modules for twisted affine Lie algebras, $q$-series multisums, and Nandi’s identities

Baker, Katherine; Kanade, Shashank; Russell, Matthew C.; Sadowski, Christopher

doi:10.5802/alco.311

Principal subspaces of basic modules for twisted affine Lie algebras,

q

-series multisums, and Nandi’s identities

Baker, Katherine ¹; Kanade, Shashank ²; Russell, Matthew C.³; Sadowski, Christopher ¹

¹ Department of Mathematics, Computer Science, and Statistics Ursinus College Collegeville PA 19426
² Department of Mathematics University of Denver Denver CO 80208
³ Department of Mathematics University of Illinois Urbana-Champaign Urbana IL 61801

Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1533-1556.

Abstract

We provide an observation relating several known and conjectured $q$ -series identities to the theory of principal subspaces of basic modules for twisted affine Lie algebras. We also state and prove two new families of $q$ -series identities. The first family provides quadruple sum representations for Nandi’s identities, including a manifestly positive representation for the first identity. The second is a family of new mod 10 identities connected with principal characters of integrable, level 4, highest-weight modules of $D_{4}^{(3)}$ .

Supplementary Materials:
Supplementary materials for this article are supplied as separate files:

Received: 2022-10-06
Revised: 2023-02-22
Accepted: 2023-04-10
Published online: 2024-01-08

DOI: 10.5802/alco.311

Classification: 05A15, 05A17, 11P84, 17B69
Keywords: Principal subspaces, vertex operator algebras, affine Lie algebras, Nandi’s identities

Author's affiliations:

Baker, Katherine ¹; Kanade, Shashank ²; Russell, Matthew C. ³; Sadowski, Christopher ¹

¹ Department of Mathematics, Computer Science, and Statistics Ursinus College Collegeville PA 19426
² Department of Mathematics University of Denver Denver CO 80208
³ Department of Mathematics University of Illinois Urbana-Champaign Urbana IL 61801

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Principal subspaces of basic modules for twisted affine {Lie} algebras, $q$-series multisums, and {Nandi{\textquoteright}s} identities},
     journal = {Algebraic Combinatorics},
     pages = {1533--1556},
     publisher = {The Combinatorics Consortium},
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Baker, Katherine; Kanade, Shashank; Russell, Matthew C.; Sadowski, Christopher. Principal subspaces of basic modules for twisted affine Lie algebras, $q$-series multisums, and Nandi’s identities. Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1533-1556. doi : 10.5802/alco.311. https://alco.centre-mersenne.org/articles/10.5802/alco.311/

References
Cited by

[1] Andrews, George E. An analytic generalization of the Rogers–Ramanujan identities for odd moduli, Proc. Nat. Acad. Sci. U.S.A., Volume 71 (1974), pp. 4082-4085 | DOI | MR | Zbl

[2] Andrews, George E. Multiple series Rogers–Ramanujan type identities, Pacific J. Math., Volume 114 (1984) no. 2, pp. 267-283 | DOI | MR | Zbl

[3] Andrews, George E. The theory of partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998, xvi+255 pages (Reprint of the 1976 original) | MR

[4] Andrews, George E.; Schilling, Anne; Warnaar, S. Ole An $A_{2}$ Bailey lemma and Rogers–Ramanujan-type identities, J. Amer. Math. Soc., Volume 12 (1999) no. 3, pp. 677-702 | DOI | MR | Zbl

[5] Bos, M. K. Coding the principal character formula for affine Kac–Moody Lie algebras, Math. Comp., Volume 72 (2003) no. 244, pp. 2001-2012 | DOI | MR | Zbl

[6] Bressoud, D.; Ismail, M. E. H.; Stanton, D. Change of base in Bailey pairs, Ramanujan J., Volume 4 (2000) no. 4, pp. 435-453 | DOI | MR | Zbl

[7] Bressoud, David M. Analytic and combinatorial generalizations of the Rogers–Ramanujan identities, Mem. Amer. Math. Soc., Volume 24 (1980) no. 227, p. 54 | DOI | MR | Zbl

[8] Bringmann, Kathrin; Jennings-Shaffer, Chris; Mahlburg, Karl Proofs and reductions of various conjectured partition identities of Kanade and Russell, J. Reine Angew. Math., Volume 766 (2020), pp. 109-135 | DOI | MR | Zbl

[9] Calinescu, C.; Lepowsky, J.; Milas, A. Vertex-algebraic structure of the principal subspaces of certain $A_{1}^{(1)}$ -modules. I. Level one case, Internat. J. Math., Volume 19 (2008) no. 1, pp. 71-92 | DOI | MR | Zbl

[10] Calinescu, C.; Lepowsky, J.; Milas, A. Vertex-algebraic structure of the principal subspaces of certain $A_{1}^{(1)}$ -modules. II. Higher-level case, J. Pure Appl. Algebra, Volume 212 (2008) no. 8, pp. 1928-1950 | DOI | MR | Zbl

[11] Calinescu, C.; Lepowsky, J.; Milas, A. Vertex-algebraic structure of the principal subspaces of level one modules for the untwisted affine Lie algebras of types $A, D, E$ , J. Algebra, Volume 323 (2010) no. 1, pp. 167-192 | DOI | MR | Zbl

[12] Calinescu, C.; Lepowsky, J.; Milas, A. Vertex-algebraic structure of principal subspaces of standard $A_{2}^{(2)}$ -modules, I, Internat. J. Math., Volume 25 (2014) no. 7, Paper no. 1450063, 44 pages | DOI | MR | Zbl

[13] Calinescu, Corina; Milas, Antun; Penn, Michael Vertex algebraic structure of principal subspaces of basic $A_{2 n}^{(2)}$ -modules, J. Pure Appl. Algebra, Volume 220 (2016) no. 5, pp. 1752-1784 | DOI | MR | Zbl

[14] Calinescu, Corina; Penn, Michael; Sadowski, Christopher Presentations of principal subspaces of higher level standard $A_{2}^{(2)}$ -modules, Algebr. Represent. Theory, Volume 22 (2019) no. 6, pp. 1457-1478 | DOI | MR | Zbl

[15] Capparelli, S.; Lepowsky, J.; Milas, A. The Rogers–Ramanujan recursion and intertwining operators, Commun. Contemp. Math., Volume 5 (2003) no. 6, pp. 947-966 | DOI | MR | Zbl

[16] Capparelli, S.; Lepowsky, J.; Milas, A. The Rogers–Selberg recursions, the Gordon–Andrews identities and intertwining operators, Ramanujan J., Volume 12 (2006) no. 3, pp. 379-397 | DOI | MR | Zbl

[17] Capparelli, Stefano On some representations of twisted affine Lie algebras and combinatorial identities, J. Algebra, Volume 154 (1993) no. 2, pp. 335-355 | DOI | MR | Zbl

[18] Carter, R. W. Lie algebras of finite and affine type, Cambridge Studies in Advanced Mathematics, 96, Cambridge University Press, Cambridge, 2005, xviii+632 pages | DOI | MR

[19] Chern, Shane Linked partition ideals, directed graphs and $q$ -multi-summations, Electron. J. Combin., Volume 27 (2020) no. 3, Paper no. 3.33, 29 pages | DOI | MR | Zbl

[20] Corteel, Sylvie; Dousse, Jehanne; Uncu, Ali Kemal Cylindric partitions and some new $A_{2}$ Rogers–Ramanujan identities, Proc. Amer. Math. Soc., Volume 150 (2022) no. 2, pp. 481-497 | DOI | MR | Zbl

[21] Feigin, Boris; Stoyanovsky, A. V. Quasi-particles models for the representations of Lie algebras and geometry of flag manifold, 1993 | arXiv

[22] Garvan, Frank A $q$ -product tutorial for a $q$ -series MAPLE package, Sém. Lothar. Combin., Volume 42 (1999), Paper no. B42d, 27 pages The Andrews Festschrift (Maratea, 1998) | MR | Zbl

[23] Griffin, Michael J.; Ono, Ken; Warnaar, S. Ole A framework of Rogers–Ramanujan identities and their arithmetic properties, Duke Math. J., Volume 165 (2016) no. 8, pp. 1475-1527 | DOI | MR | Zbl

[24] Kanade, Shashank; Russell, Matthew C. IdentityFinder and some new identities of Rogers–Ramanujan type, Exp. Math., Volume 24 (2015) no. 4, pp. 419-423 | DOI | MR | Zbl

[25] Kanade, Shashank; Russell, Matthew C. Staircases to analytic sum-sides for many new integer partition identities of Rogers–Ramanujan type, Electron. J. Combin., Volume 26 (2019) no. 1, Paper no. 1.6, 33 pages | DOI | MR | Zbl

[26] Kanade, Shashank; Russell, Matthew C. On $q$ -series for principal characters of standard $A_{2}^{(2)}$ -modules, Adv. Math., Volume 400 (2022), Paper no. 108282, 24 pages | DOI | MR | Zbl

[27] Kanade, Shashank; Russell, Matthew C. Completing the A2 Andrews–Schilling–Warnaar Identities, Int. Math. Res. Not. IMRN (2023) no. 20, pp. 17100-17155 | DOI

[28] Konenkov, Stepan Further $q$ -reflections on the modulo 9 Kanade–Russell (conjectural) identities, 2022 | arXiv

[29] Kurşungöz, Kağan Andrews–Gordon type series for Capparelli’s and Göllnitz–Gordon identities, J. Combin. Theory Ser. A, Volume 165 (2019), pp. 117-138 | DOI | MR | Zbl

[30] Kurşungöz, Kağan Andrews–Gordon type series for Kanade–Russell conjectures, Ann. Comb., Volume 23 (2019) no. 3-4, pp. 835-888 | DOI | MR | Zbl

[31] Lepowsky, J. Calculus of twisted vertex operators, Proc. Nat. Acad. Sci. U.S.A., Volume 82 (1985) no. 24, pp. 8295-8299 | DOI | MR | Zbl

[32] Lepowsky, James; Li, Haisheng Introduction to vertex operator algebras and their representations, Progress in Mathematics, 227, Birkhäuser Boston, Inc., Boston, MA, 2004, xiv+318 pages | DOI | MR

[33] Nandi, Debajyoti Partition identities arising from the standard A2 (2)-modules of level 4, ProQuest LLC, Ann Arbor, MI, 2014, 203 pages Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick | MR

[34] Paule, Peter On identities of the Rogers–Ramanujan type, J. Math. Anal. Appl., Volume 107 (1985) no. 1, pp. 255-284 | DOI | MR | Zbl

[35] Penn, Michael; Sadowski, Christopher Vertex-algebraic structure of principal subspaces of basic $D_{4}^{(3)}$ -modules, Ramanujan J., Volume 43 (2017) no. 3, pp. 571-617 | DOI | MR | Zbl

[36] Penn, Michael; Sadowski, Christopher Vertex-algebraic structure of principal subspaces of the basic modules for twisted affine Lie algebras of type $A_{2 n - 1}^{(2)}$ , $D_{n}^{(2)}$ , $E_{6}^{(2)}$ , J. Algebra, Volume 496 (2018), pp. 242-291 | DOI | MR | Zbl

[37] Penn, Michael; Sadowski, Christopher; Webb, Gautam Principal subspaces of twisted modules for certain lattice vertex operator algebras, Internat. J. Math., Volume 30 (2019) no. 10, Paper no. 1950048, 47 pages | DOI | MR | Zbl

[38] Rosengren, Hjalmar Proofs of some partition identities conjectured by Kanade and Russell, Ramanujan J., Volume 61 (2023) no. 1, pp. 295-317 | DOI | MR | Zbl

[39] Russell, Matthew Christopher Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences, ProQuest LLC, Ann Arbor, MI, 2016, 74 pages Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick | MR

[40] Sills, Andrew V. An invitation to the Rogers–Ramanujan identities, CRC Press, Boca Raton, FL, 2018, xx+233 pages (With a foreword by George E. Andrews) | MR

[41] Stembridge, John R. Hall–Littlewood functions, plane partitions, and the Rogers–Ramanujan identities, Trans. Amer. Math. Soc., Volume 319 (1990) no. 2, pp. 469-498 | DOI | MR | Zbl

[42] Stoyanovskiĭ, A. V.; Feĭgin, B. L. Functional models of the representations of current algebras, and semi-infinite Schubert cells, Funktsional. Anal. i Prilozhen., Volume 28 (1994) no. 1, p. 68-90, 96 | DOI | MR

[43] Takenaka, Ryo Vertex algebraic construction of modules for twisted affine Lie algebras of type $A_{2 l}^{(2)}$ , J. Pure Appl. Algebra, Volume 227 (2023) no. 4, Paper no. 107263, 33 pages | MR | Zbl

[44] Takigiku, Motoki; Tsuchioka, Shunsuke A proof of conjectured partition identities of Nandi, 2020 (Forthcoming, Amer. J. Math) | arXiv

[45] Takigiku, Motoki; Tsuchioka, Shunsuke Andrews–Gordon type series for the level 5 and 7 standard modules of the affine Lie algebra $A_{2}^{(2)}$ , Proc. Amer. Math. Soc., Volume 149 (2021) no. 7, pp. 2763-2776 | DOI | MR | Zbl

[46] Tsuchioka, Shunsuke An example of $A_{2}$ Rogers–Ramanujan bipartition identities of level 3, 2022 | arXiv

[47] Tsuchioka, Shunsuke A vertex operator reformulation of the Kanade–Russell conjecture modulo 9, 2022 | arXiv

[48] Uncu, Ali; Zudilin, Wadim Reflecting (on) the modulo 9 Kanade–Russell (conjectural) identities, Sém. Lothar. Combin., Volume 85 ([2020–2021]), Paper no. B85e, 17 pages | MR | Zbl

[49] Warnaar, S. Ole The generalized Borwein conjecture. II. Refined $q$ -trinomial coefficients, Discrete Math., Volume 272 (2003) no. 2-3, pp. 215-258 | DOI | MR | Zbl

[50] Warnaar, S. Ole The $A_{2}$ Andrews-Gordon identities and cylindric partitions, Trans. Amer. Math. Soc. Ser. B, Volume 10 (2023), pp. 715-765 | DOI | MR | Zbl

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