ALGEBRAIC COMBINATORICS

no. 3
Equivariant quantum cohomology of the Grassmannian via the rim hook rule
Bertiger, Anna1; Milićević, Elizabeth2; Taipale, Kaisa3
1 Microsoft, Seattle, WA, USA
2 Department of Mathematics & Statistics, Haverford College, Haverford, PA, USA
3 Department of Mathematics, University of Minnesota, Minneapolis, MN, USA
Algebraic Combinatorics, Volume 1 (2018) no. 3, pp. 327-352.

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 k-planes in complex n-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 n, suggesting a direct connection to the Peterson isomorphism relating quantum and affine Schubert calculus.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.14
Classification: 14N35,  14N15,  14M15,  55N91,  05E05
Keywords: Schubert calculus, equivariant quantum cohomology, core partition, abacus diagram, factorial Schur polynomial
Bertiger, Anna 1; Milićević, Elizabeth 2; Taipale, Kaisa 3

1 Microsoft, Seattle, WA, USA
2 Department of Mathematics & Statistics, Haverford College, Haverford, PA, USA
3 Department of Mathematics, University of Minnesota, Minneapolis, MN, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@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 = {https://alco.centre-mersenne.org/articles/10.5802/alco.14/}
}
TY  - JOUR
TI  - Equivariant quantum cohomology of the Grassmannian via the rim hook rule
JO  - Algebraic Combinatorics
PY  - 2018
DA  - 2018///
SP  - 327
EP  - 352
VL  - 1
IS  - 3
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.14/
UR  - https://zbmath.org/?q=an%3A06897704
UR  - https://www.ams.org/mathscinet-getitem?mr=3856527
UR  - https://doi.org/10.5802/alco.14
DO  - 10.5802/alco.14
LA  - en
ID  - ALCO_2018__1_3_327_0
ER  - 
%0 Journal Article
%T Equivariant quantum cohomology of the Grassmannian via the rim hook rule
%J Algebraic Combinatorics
%D 2018
%P 327-352
%V 1
%N 3
%I MathOA foundation
%U https://doi.org/10.5802/alco.14
%R 10.5802/alco.14
%G en
%F 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/articles/10.5802/alco.14/

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

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

[3] Buch, Anders Skovsted Mutations of puzzles and equivariant cohomology of two-step flag varieties, Ann. Math., Volume 182 (2015) no. 1, pp. 173-220 | DOI | MR | Zbl

[4] Buch, Anders Skovsted; Kresch, Andrew; Purbhoo, Kevin; Tamvakis, Harry The puzzle conjecture for the cohomology of two-step flag manifolds, J. Algebr. Comb., Volume 44 (2016) no. 4, pp. 973-1007 | DOI | MR | Zbl

[5] Buch, Anders Skovsted; Kresch, Andrew; Tamvakis, Harry Gromov-Witten invariants on Grassmannians, J. Am. Math. Soc., Volume 16 (2003) no. 4, pp. 901-915 | DOI | MR | Zbl

[6] Buch, Anders Skovsted; Mihalcea, Leonardo Constantin Quantum K-theory of Grassmannians, Duke Math. J., Volume 156 (2011) no. 3, pp. 501-538 | DOI | MR | Zbl

[7] Chen, William Y. C.; Louck, James D. The factorial Schur function, J. Math. Phys., Volume 34 (1993) no. 9, pp. 4144-4160 | DOI | MR | Zbl

[8] Fulton, William Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, 1997, x+260 pages | MR | Zbl

[9] Fulton, William; Pandharipande, Rahul 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 | DOI | MR | Zbl

[10] Givental, Alexander B. Equivariant Gromov-Witten invariants, Int. Math. Res. Not. (1996) no. 13, pp. 613-663 | DOI | MR | Zbl

[11] Givental, Alexander B.; Kim, Bumsig Quantum cohomology of flag manifolds and Toda lattices, Commun. Math. Phys., Volume 168 (1995) no. 3, pp. 609-641 | DOI | MR | Zbl

[12] Gorbounov, Vassily; Korff, Christian Equivariant quantum cohomology and Yang-Baxter algebras (2014) (https://arxiv.org/abs/1402.2907)

[13] Graham, William Positivity in equivariant Schubert calculus, Duke Math. J., Volume 109 (2001) no. 3, pp. 599-614 | DOI | MR | Zbl

[14] 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, xxviii+510 pages (With a foreword by P. M. Cohn and an introduction by Gilbert de B. Robinson) | MR | Zbl

[15] Knutson, Allen; Tao, Terence Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J., Volume 119 (2003) no. 2, pp. 221-260 | DOI | MR | Zbl

[16] Laksov, Dan A formalism for equivariant Schubert calculus, Algebra Number Theory, Volume 3 (2009) no. 6, pp. 711-727 | DOI | MR | Zbl

[17] Lam, Thomas; Shimozono, Mark Quantum cohomology of G/P and homology of affine Grassmannian, Acta Math., Volume 204 (2010) no. 1, pp. 49-90 | DOI | MR | Zbl

[18] Lam, Thomas; Shimozono, Mark From double quantum Schubert polynomials to k-double Schur functions via the Toda lattice (2011) (https://arxiv.org/abs/1109.2193) | Zbl

[19] Lam, Thomas; Shimozono, Mark k-double Schur functions and equivariant (co)homology of the affine Grassmannian, Math. Ann., Volume 356 (2013) no. 4, pp. 1379-1404 | DOI | MR | Zbl

[20] Li, Changzheng; Ravikumar, Vijay Equivariant Pieri rules for isotropic Grassmannians, Math. Ann., Volume 365 (2016) no. 1-2, pp. 881-909 | DOI | MR | Zbl

[21] Macdonald, Ian Grant 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

[22] Mihalcea, Leonardo Constantin Equivariant quantum Schubert calculus, Adv. Math., Volume 203 (2006) no. 1, pp. 1-33 | DOI | MR | Zbl

[23] Mihalcea, Leonardo Constantin Giambelli formulae for the equivariant quantum cohomology of the Grassmannian, Trans. Am. Math. Soc., Volume 360 (2008) no. 5, pp. 2285-2301 | DOI | MR | Zbl

[24] Molev, Alexander I.; Sagan, Bruce E. A Littlewood-Richardson rule for factorial Schur functions, Trans. Am. Math. Soc., Volume 351 (1999) no. 11, pp. 4429-4443 | DOI | MR | Zbl

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

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

[27] Santiago, Taíse Schubert calculus on a Grassmann algebra (2006) (Ph. D. Thesis)

[28] The Sage Developers SageMath, the Sage Mathematics Software System (Version 5.10), 2013 (http://www.sagemath.org/)

Cited by Sources: