A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provided a way to compute quantum products of Schubert classes in the Grassmannian of -planes in complex -space by doing classical multiplication and then applying a combinatorial rim hook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao’s puzzle rule then gives an effective algorithm for computing all equivariant quantum Littlewood–Richardson coefficients. Interestingly, this rule requires a specialization of torus weights modulo , suggesting a direct connection to the Peterson isomorphism relating quantum and affine Schubert calculus.
Revised: 2018-01-21
Accepted: 2018-02-08
Published online: 2018-06-28
DOI: https://doi.org/10.5802/alco.14
Classification: 14N35, 14N15, 14M15, 55N91, 05E05
Keywords: Schubert calculus, equivariant quantum cohomology, core partition, abacus diagram, factorial Schur polynomial
@article{ALCO_2018__1_3_327_0, author = {Bertiger, Anna and Mili\'cevi\'c, Elizabeth and Taipale, Kaisa}, title = {Equivariant quantum cohomology of the Grassmannian via the rim hook rule}, journal = {Algebraic Combinatorics}, pages = {327--352}, publisher = {MathOA foundation}, volume = {1}, number = {3}, year = {2018}, doi = {10.5802/alco.14}, zbl = {06897704}, mrnumber = {3856527}, language = {en}, url = {alco.centre-mersenne.org/item/ALCO_2018__1_3_327_0/} }
Bertiger, Anna; Milićević, Elizabeth; Taipale, Kaisa. Equivariant quantum cohomology of the Grassmannian via the rim hook rule. Algebraic Combinatorics, Volume 1 (2018) no. 3, pp. 327-352. doi : 10.5802/alco.14. https://alco.centre-mersenne.org/item/ALCO_2018__1_3_327_0/
[1] Quantum Schubert calculus, Adv. Math., Volume 128 (1997) no. 2, pp. 289-305 | Article | MR 1454400 | Zbl 0945.14031
[2] Quantum multiplication of Schur polynomials, J. Algebra, Volume 219 (1999) no. 2, pp. 728-746 | Article | MR 1706853 | Zbl 0936.05086
[3] Mutations of puzzles and equivariant cohomology of two-step flag varieties, Ann. Math., Volume 182 (2015) no. 1, pp. 173-220 | Article | MR 3374959 | Zbl 1354.14072
[4] The puzzle conjecture for the cohomology of two-step flag manifolds, J. Algebr. Comb., Volume 44 (2016) no. 4, pp. 973-1007 | Article | MR 3566227 | Zbl 1356.05154
[5] Gromov-Witten invariants on Grassmannians, J. Am. Math. Soc., Volume 16 (2003) no. 4, pp. 901-915 | Article | MR 1992829 | Zbl 1063.53090
[6] Quantum -theory of Grassmannians, Duke Math. J., Volume 156 (2011) no. 3, pp. 501-538 | Article | MR 2772069 | Zbl 1213.14103
[7] The factorial Schur function, J. Math. Phys., Volume 34 (1993) no. 9, pp. 4144-4160 | Article | MR 1233264 | Zbl 0787.05091
[8] Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, Volume 35, Cambridge University Press, 1997, x+260 pages | MR 1464693 | Zbl 0878.14034
[9] Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995 (Proceedings of Symposia in Pure Mathematics) Volume 62, American Mathematical Society, Providence, RI, 1997, pp. 45-96 | Article | MR 1492534 | Zbl 0898.14018
[10] Equivariant Gromov-Witten invariants, Int. Math. Res. Not. (1996) no. 13, pp. 613-663 | Article | MR 1408320 | Zbl 0881.55006
[11] Quantum cohomology of flag manifolds and Toda lattices, Commun. Math. Phys., Volume 168 (1995) no. 3, pp. 609-641 | Article | MR 1328256 | Zbl 0828.55004
[12] Equivariant quantum cohomology and Yang-Baxter algebras (2014) (https://arxiv.org/abs/1402.2907)
[13] Positivity in equivariant Schubert calculus, Duke Math. J., Volume 109 (2001) no. 3, pp. 599-614 | Article | MR 1853356 | Zbl 1069.14055
[14] The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, Volume 16, Addison-Wesley Publishing Co., Reading, Mass., 1981, xxviii+510 pages (With a foreword by P. M. Cohn and an introduction by Gilbert de B. Robinson) | MR 644144 | Zbl 1981
[15] Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J., Volume 119 (2003) no. 2, pp. 221-260 | Article | MR 1997946 | Zbl 1064.14063
[16] A formalism for equivariant Schubert calculus, Algebra Number Theory, Volume 3 (2009) no. 6, pp. 711-727 | Article | MR 2579392 | Zbl 1186.14057
[17] Quantum cohomology of and homology of affine Grassmannian, Acta Math., Volume 204 (2010) no. 1, pp. 49-90 | Article | MR 2600433 | Zbl 1216.14052
[18] From double quantum Schubert polynomials to -double Schur functions via the Toda lattice (2011) (https://arxiv.org/abs/1109.2193) | Zbl 1269.05113
[19] -double Schur functions and equivariant (co)homology of the affine Grassmannian, Math. Ann., Volume 356 (2013) no. 4, pp. 1379-1404 | Article | MR 3072805 | Zbl 1282.14092
[20] Equivariant Pieri rules for isotropic Grassmannians, Math. Ann., Volume 365 (2016) no. 1-2, pp. 881-909 | Article | MR 3498930 | Zbl 1339.14030
[21] Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995, x+475 pages (With contributions by A. Zelevinsky, Oxford Science Publications) | Zbl 0824.05059
[22] Equivariant quantum Schubert calculus, Adv. Math., Volume 203 (2006) no. 1, pp. 1-33 | Article | MR 2231042 | Zbl 1100.14045
[23] Giambelli formulae for the equivariant quantum cohomology of the Grassmannian, Trans. Am. Math. Soc., Volume 360 (2008) no. 5, pp. 2285-2301 | Article | MR 2373314 | Zbl 1136.14046
[24] A Littlewood-Richardson rule for factorial Schur functions, Trans. Am. Math. Soc., Volume 351 (1999) no. 11, pp. 4429-4443 | Article | MR 1621694 | Zbl 0972.05053
[25] Affine approach to quantum Schubert calculus, Duke Math. J., Volume 128 (2005) no. 3, pp. 473-509 | Article | MR 2145741 | Zbl 1081.14070
[26] Quantum cohomology rings of Grassmannians and total positivity, Duke Math. J., Volume 110 (2001) no. 3, pp. 523-553 | Article | MR 1869115 | Zbl 1013.14014
[27] Schubert calculus on a Grassmann algebra (2006) (Ph. D. Thesis)
[28] SageMath, the Sage Mathematics Software System (Version 5.10), 2013 (http://www.sagemath.org/)