Cylindric symmetric functions and positivity

Korff, Christian¹ ; Palazzo, David¹

¹ School of Mathematics and Statistics University of Glasgow Glasgow G12 8QQ, UK

Algebraic Combinatorics, Tome 3 (2020) no. 1, pp. 191-247.

Résumé

We introduce new families of cylindric symmetric functions as subcoalgebras in the ring of symmetric functions $\Lambda$ (viewed as a Hopf algebra) which have non-negative structure constants. Combinatorially these cylindric symmetric functions are defined as weighted sums over cylindric reverse plane partitions or - alternatively - in terms of sets of affine permutations. We relate their combinatorial definition to an algebraic construction in terms of the principal Heisenberg subalgebra of the affine Lie algebra ${\widehat{\mathrm{𝔰𝔩}}}_n$ and a specialised cyclotomic Hecke algebra. Using Schur–Weyl duality we show that the new cylindric symmetric functions arise as matrix elements of Lie algebra elements in the subspace of symmetric tensors of a particular level-0 module which can be identified with the small quantum cohomology ring of the $k$-fold product of projective space. The analogous construction in the subspace of alternating tensors gives the known set of cylindric Schur functions which are related to the small quantum cohomology ring of Grassmannians. We prove that cylindric Schur functions form a subcoalgebra in $\Lambda$ whose structure constants are the 3-point genus 0 Gromov–Witten invariants. We show that the new families of cylindric functions obtained from the subspace of symmetric tensors also share the structure constants of a symmetric Frobenius algebra, which we define in terms of tensor multiplicities of the generalised symmetric group $G(n,1,k)$.

Détail

Reçu le : 2018-05-01
Révisé le : 2019-07-03
Accepté le : 2019-07-03
Publié le : 2020-02-12

DOI : 10.5802/alco.90

Classification : 05E05, 05E10, 14N35, 53D45
Mots-clés : Cylindric reverse plane partitions, symmetric functions, 2D TQFT, Gromov–Witten invariants

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Bibliographie

[1] Abrams, Lowell The quantum Euler class and the quantum cohomology of the Grassmannians, Isr. J. Math., Volume 117 (2000) no. 1, pp. 335-352 | MR | DOI | Zbl

[2] Agnihotri, Sharad Quantum cohomology and the Verlinde algebra, Ph. D. Thesis, University of Oxford (United Kingdom) (1995)

[3] Ariki, Susumu On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ., Volume 36 (1996) no. 4, pp. 789-808 | DOI | MR | Zbl

[4] Ariki, Susumu; Koike, Kazuhiko A Hecke algebra of $(Z/rZ)\wr S_n$ and construction of its irreducible representations, Adv. Math, Volume 106 (1994) no. 2, pp. 216-243 | Zbl | DOI

[5] Atiyah, Michael F. Topological quantum field theories, Math. Publ. IHES, Volume 68 (1988), pp. 175-186 | mathdoc-id | DOI | Zbl

[6] Beauville, Arnaud Conformal Blocks, Fusion Rules, and the Verlinde formula, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Israel Math. Conf. Proc.), Volume 9, Weizmann Science Press of Israel (1996), pp. 75-96 | MR | Zbl

[7] Bertram, Aaron Quantum Schubert calculus, Adv. Math, Volume 128 (1997) no. 2, pp. 289-305 | MR | DOI | Zbl

[8] Bertram, Aaron; Ciocan-Fontanine, Ionuţ; Fulton, William Quantum multiplication of Schur polynomials, J. Algebra, Volume 219 (1999) no. 2, pp. 728-746 | MR | DOI | Zbl

[9] Bertram, Aaron; Ciocan-Fontanine, Ionuţ; Kim, Bumsig Two proofs of a conjecture of Hori and Vafa, Duke Math. J., Volume 126 (2005) no. 1, pp. 101-136 | Zbl | MR | DOI

[10] Björner, Anders; Brenti, Francesco Affine permutations of type A, Electron. J. Combin, Volume 3 (1996) no. 2, R18, 35 pages | MR | Zbl

[11] Butler, Lynne M.; Hales, Alfred W. Nonnegative Hall polynomials, J. Algebr. Comb., Volume 2 (1993) no. 2, pp. 125-135 | MR | DOI | Zbl

[12] Eriksson, Henrik; Eriksson, Kimmo Affine Weyl groups as infinite permutations, Electron. J. Combin, Volume 5 (1998) no. 1, R18, 35 pages | MR | Zbl

[13] Faltings, Gerd A proof for the Verlinde formula, Journal of Algebraic Geometry, Volume 3 (1994) no. 2, pp. 347-374 | MR | Zbl

[14] Di Francesco, Philippe; Mathieu, Pierre; Sénéchal, David Conformal field theory, Springer Science & Business Media, 1997 | Zbl

[15] Freed, Daniel S.; Hopkins, Michael J.; Teleman, Constantin Loop groups and twisted K-theory I, J. Topol., Volume 4 (2011) no. 4, pp. 737-798 | MR | DOI | Zbl

[16] Freed, Daniel S.; Hopkins, Michael J.; Teleman, Constantin Loop groups and twisted K-theory II, J. Am. Math. Soc., Volume 26 (2013) no. 3, pp. 595-644 | DOI | MR | Zbl

[17] Fulton, William Young tableaux: with applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, 1997 | Zbl

[18] Fulton, William; Pandharipande, Rahul Notes on stable maps and quantum cohomology, Algebraic Geometry Santa Cruz 1995 (Proceedings of Symposia in Pure Mathematics), Volume 62, Amer. Math. Soc. (1997) no. 2, pp. 45-96 | Zbl | MR | DOI

[19] Gepner, Doron Fusion rings and geometry, Comm. Math. Phys., Volume 141 (1991) no. 2, pp. 381-411 | MR | DOI | Zbl

[20] Gessel, Ira; Krattenthaler, Christian Cylindric partitions, Trans. Am. Math. Soc., Volume 349 (1997) no. 2, pp. 429-479 | MR | DOI | Zbl

[21] Ginzburg, Victor Perverse sheaves on a loop group and Langlands’ duality (1995) (preprint https://arxiv.org/abs/alg-geom/9511007)

[22] Golyshev, Vasily; Manivel, Laurent Quantum cohomology and the Satake isomorphism (2011) (https://arxiv.org/abs/1106.3120)

[23] Hori, Kentaro; Vafa, Cumrun Mirror symmetry (2000) (https://arxiv.org/abs/hep-th/0002222)

[24] Intriligator, Kenneth Fusion residues, Mod. Phys. Lett. A, Volume 6 (1991) no. 38, pp. 3543-3556 | MR | DOI | Zbl

[25] James, Gordon; Kerber, Adalbert The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, Mass, 1981 | MR | Zbl

[26] Kac, Victor G. Infinite-dimensional Lie algebras, Cambridge University Press, 1994 | Zbl

[27] Kac, Victor G.; Peterson, Dale H. Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math, Volume 53 (1984) no. 2, pp. 125-264 | MR | Zbl

[28] Kim, Bumsig; Sabbah, Claude Quantum cohomology of the Grassmannian and alternate Thom–Sebastiani, Compos. Math., Volume 144 (2008) no. 1, pp. 221-246 | MR | Zbl

[29] Kock, Joachim Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts, 59, Cambridge University Press, 2004 | MR | Zbl

[30] Korff, Christian Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra, Commun. Math. Phys., Volume 318 (2013) no. 1, pp. 173-246 | MR | DOI | Zbl

[31] Korff, Christian; Stroppel, Catharina The $\widehat{sl}{\left(n\right)}_k$-WZNW fusion ring: a combinatorial construction and a realisation as quotient of quantum cohomology, Adv. Math, Volume 225 (2010) no. 1, pp. 200-268 | MR | DOI | Zbl

[32] Kostant, Bertram The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group, Am. J. Math., Volume 81 (1959) no. 4, pp. 973-1032 | MR | DOI | Zbl

[33] Lam, Thomas Affine Stanley symmetric functions, Am. J. Math., Volume 128 (2006) no. 6, pp. 1553-1586 | MR | Zbl

[34] Lee, Seung Jin Positivity of cylindric skew Schur functions, J. Comb. Theory, Ser. A, Volume 168 (2019), pp. 26-49 | MR | Zbl

[35] Lusztig, George Some examples of square integrable representations of semisimple $p$-adic groups, Trans. Am. Math. Soc., Volume 277 (1983) no. 2, pp. 623-653 | MR | Zbl

[36] Macdonald, Ian Grant Symmetric functions and Hall polynomials, Oxford University Press, 1998 | Zbl

[37] McNamara, Peter Cylindric skew Schur functions, Adv. Math, Volume 205 (2006) no. 1, pp. 275-312 | MR | DOI | Zbl

[38] Morrison, Andrew; Sottile, Frank Two Murnaghan–Nakayama Rules in Schubert Calculus, Ann. Comb., Volume 22 (2018) no. 2, pp. 363-375 | MR | DOI | Zbl

[39] Naculich, Stephen G.; Schnitzer, Howard J. Level-rank duality of the $U\left(N\right)$ WZW model, Chern–Simons theory, and 2d qYM theory, J. High Energy Phys. (2007), JHEP06(2007)023, 20 pages | DOI

[40] Osima, Masaru On the representations of the generalized symmetric group, Math. J. Okayama Univ, Volume 4 (1954) no. 1, pp. 39-56 | MR | Zbl

[41] Postnikov, Alexander Affine approach to quantum Schubert calculus, Duke Math. J., Volume 128 (2005) no. 3, pp. 473-509 | MR | DOI | Zbl

[42] Pushkarev, Igor A. On the representation theory of wreath products of finite groups and symmetric groups, J. Math. Sci., Volume 96 (1999) no. 5, pp. 3590-3599 | DOI

[43] Rietsch, Konstanze Quantum cohomology rings of Grassmannians and total positivity, Duke Math. J., Volume 110 (2001) no. 3, pp. 523-553 | MR | DOI | Zbl

[44] Shi, Jian-yi The Kazhdan–Lusztig cells in certain affine Weyl groups, Lecture Notes in Math, 1179, Springer, 1986 | MR | Zbl

[45] Siebert, Bernd; Tian, Gang On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math, Volume 1 (1997) no. 4, pp. 679-695 | MR | DOI | Zbl

[46] Specht, Wilhelm Eine Verallgemeinerung der symmetrischen Gruppe, Ph. D. Thesis, Humboldt-Universität zu Berlin (Germany) (1932) | Zbl | DOI

[47] Stanley, Richard P. Enumerative Combinatorics, Volume 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999 | Zbl

[48] Teleman, Constantin K-theory of the moduli space of bundles on a surface and deformations of the Verlinde algebra, Topology, geometry and quantum field theory. Proceedings of the 2002 Oxford symposium in honour of the 60th birthday of Graeme Segal, Oxford, UK, June 24–29, 2002 (London Mathematical Society Lecture Notes Series), Volume 308, Cambridge University Press (2004), pp. 358-378 | MR | Zbl

[49] Teleman, Constantin; Woodward, Christopher T. The index formula for the moduli of G-bundles on a curve, Annals of Math., Volume 170 (2009) no. 2, pp. 495-527 | MR | DOI | Zbl

[50] Vafa, Cumrun Topological mirrors and quantum rings, Essays on Mirror Manifolds, International Press, 1992, pp. 96-117 | Zbl

[51] Verlinde, Erik Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys., B, Volume 300 (1988), pp. 360-376 | DOI | Zbl

[52] Witten, Edward The Verlinde algebra and the cohomology of the Grassmannian, Geometry, topology and physics, Conf. Proc. Lecture Notes Geom. Topology, IV (1995), pp. 357-422 | Zbl

[53] Zelevinsky, Andrei V. Representations of Finite Classical Groups: A Hopf Algebra Approach, Lecture Notes in Math, 869, Springer, 1981 | Zbl