Abstract
J. De Loera & T. McAllister and K. D. Mulmuley & H. Narayanan & M. Sohoni independently proved that determining the vanishing of Littlewood–Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood–Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas, A. Yong. Our proof then combines a saturation theorem of D. Anderson, E. Richmond, A. Yong, a reading order independence property, and É. Tardos’ algorithm for combinatorial linear programming.
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Anderson, D., Richmond, E., Yong, A.: Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians. Compositio Math. 149, 1569–1582 (2013)
Belkale, P.: Quantum generalization of the Horn conjecture. J. Amer. Math. Soc. 21(2), 365–408 (2008)
Buch, A.S.: A Littlewood-Richardson rule for the \(K\)-theory of Grassmannians. Acta Math. 189(1), 37–78 (2002)
Buch, A.S., Kresch, A., Purbhoo, K., Tamvakis, H.: The puzzle conjecture for the cohomology of two-step flag manifolds. J. Algebraic Combin. 44(4), 973–1007 (2016)
Buch, A.S., Kresch, A., Tamvakis, H.: Gromov-Witten invariants on Grassmannians. J. Amer. Math. Soc. 16, 901–915 (2003)
P. Bürgisser & C. Ikenmeyer (2008). The complexity of computing Kronecker coefficients. Discrete Mathematics and Theoretical Computer Science, Proc., Assoc. DMTCS, Nancy 357–368.
De Loera, J.A., McAllister, T.B.: On the computation of Clebsch-Gordan coefficients and the dilation effect. Experiment. Math. 15(1), 7–19 (2006)
Graham, W.: Positivity in equivariant Schubert calculus. Duke Math. J. 109(3), 599–614 (2001)
M. Grotschel, L. Lovasz & A. Schrijver (1993). Geometric algorithms and combinatorial optimization. Springer Verlag.
Ikenmeyer, C., Mulmuley, K.D., Walter, M.: On vanishing of Kronecker coefficients. Computational complexity 26(2), 949–992 (2017)
Knutson, A., Miller, E., Yong, A.: Gröbner geometry of vertex decompositions and of flagged tableaux. J. Reine Angew. Math. (Crelles J.) 630, 1–31 (2009)
Knutson, A., Tao, T.: Puzzles and (equivariant) cohomology of Grassmannians. Duke Math. J. 119(2), 221–260 (2003)
Kreiman, V.: Equivariant Littlewood-Richardson skew tableaux. Trans. Amer. Math. Soc. 362(5), 2589–2617 (2010)
I. G. Macdonald (1992). Schur functions: theme and variations. Actes 28-e Séminaire Lotharingien, I.R.M.A., Strasbourg 5–39.
L. Manivel (2001). Symmetric functions, Schubert polynomials and degeneracy loci. Translated from the 1998 French original by John R. Swallow. SMF/AMS Texts and Monographs, American Mathematical Society, Providence.
Molev, A.: Littlewood-Richardson polynomials. J. Algebra 321(11), 3450–3468 (2009)
Molev, A., Sagan, B.: A Littlewood-Richardson rule for factorial Schur functions. Trans. Amer. Math. Soc. 351(11), 4429–4443 (1999)
Mulmuley, K.D., Narayanan, H., Sohoni, M.: Geometric complexity theory III: on deciding nonvanishing of a Littlewood-Richardson coefficient. J. Algebraic Combin. 36(1), 103–110 (2012)
Narayanan, H.: On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients. J. Alg. Comb. 24(3), 347–354 (2006)
Pak, I., Panova, G.: On the complexity of computing Kronecker coefficients. Comput. Complexity 26(1), 1–36 (2017)
Pechenik, O., Yong, A.: Genomic tableaux. J. Algebraic Combin. 45(3), 649–685 (2017)
Tardos, É.: A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs. Operations Research 34(2), 250–256 (1986)
Thomas, H., Yong, A.: Equivariant Schubert calculus and jeu de taquin. Annales de l'Institut Fourier 68(1), 275–318 (2018)
Valiant, L.G.: The complexity of computing the permanent. Theoret. Comput. Sci. 8(2), 189–201 (1979)
Zinn-Justin, P.: Littlewood-Richardson Coefficients and Integrable Tilings. Electronic Journal of Combinatorics 16, 2131–2144 (2009)
Acknowledgements
We thank Christian Ikenmeyer, Alejandro Morales, and Igor Pak for helpful exchanges. AY was supported by an NSF grant. This material is based upon work of CR supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1746047.
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Adve, A., Robichaux, C. & Yong, A. Vanishing of Littlewood–Richardson polynomials is in P. comput. complex. 28, 241–257 (2019). https://doi.org/10.1007/s00037-019-00183-6
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DOI: https://doi.org/10.1007/s00037-019-00183-6