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Connecting Marginally Large Tableaux and Rigged Configurations via Crystals

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Abstract

We show that the bijection from rigged configurations to tensor products of Kirillov-Reshetikhin crystals extends to a crystal isomorphism between the \(B(\infty )\) models given by rigged configurations and marginally large tableaux.

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References

  1. Brubaker, B., Bump, D., Friedberg, S.: Weyl group multiple Dirichlet series, Eisenstein series and crystal bases. Ann. of Math. (2) 173(1), 1081–1120 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brubaker, B., Bump, D., Friedberg, S.: Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory, Annals of Mathematics Studies, vol. AM-175. Princeton Univ Press, New Jersey (2011)

  3. Bump, D., Nakasuji, M.: Integration on p-adic groups and crystal bases. Proc. Amer. Math. Soc 138(5), 1595–1605 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berenstein, A., Zelevinsky, A.: Canonical bases for the quantum group of type A r and piecewise-linear combinatorics. Duke Math. J 82(3), 473–502 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berenstein, A., Zelevinsky, A.: Tensor product multiplicities, canonical bases, and totally positive varieties. Invent. Math 143, 77–128 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chari, V.: On the fermionic formula and the Kirillov-Reshetikhin conjecture. Internat. Math. Res. Notices 12, 629–654 (2001). MR 1836791 (2002i:17019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cliff, G.: Crystal bases and Young tableaux. J. Algebra 202(1), 10–35 (1998). MR 1614241 (99k:17025)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deka, L., Schilling, A.: New fermionic formula for unrestricted Kostka polynomials. J. Combin. Theory Ser. A 113(7), 1435–1461 (2006). MR 2259070 (2008g:05219)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fourier, G., Okado, M., Schilling, A.: Kirillov-Reshetikhin crystals for nonexceptional types. Adv. Math. 222(3), 1080–1116 (2009). MR 2553378 (2010j:17028)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hernandez, D.: The Kirillov-Reshetikhin conjecture and solutions of T-systems. J. Reine Angew. Math. 596, 63–87 (2006). MR 2254805 (2007j:17020)

    MathSciNet  MATH  Google Scholar 

  11. Hernandez, D.: Kirillov-Reshetikhin conjecture: the general case. Int. Math. Res. Not. IMRN 1, 149–193 (2010). MR 2576287 (2011c:17029)

    MathSciNet  MATH  Google Scholar 

  12. Hong, J., Kang, S.-J.: Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, MR 1881971 (2002m:17012)

  13. Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Yamada, Y.: Remarks on fermionic formula, Recent developments in quantum affine algebras and related topics (Raleigh, NC). Contemp. Math. 248 (1998). Amer. Math. Soc. Providence, RI, 1999, pp. 243–291. MR1745263 (2001m:81129)

  14. Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Tsuboi, Z.: Paths, crystals and fermionic formulae. Math Phys odyssey.Prog. Math. Phys. 23, 205–272 (2001). Birkhäuser Boston, Boston, MA, 2002. MR1903978 (2003e:17020)

    MathSciNet  MATH  Google Scholar 

  15. Hong, J., Lee, H.: Young tableaux and crystal \(\mathcal {B(\infty )}\) for finite simple Lie algebras. J. Algebra 320(10), 3680–3693 (2008). MR 2457716 (2009j:17008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hong, J., Lee, H.: Young tableaux and crystal \(\mathcal {B(\infty )}\) for the exceptional Lie algebra types. J. Combin. Theory Ser. A 119(2), 397–419 (2012). MR 2860601 (2012i:17012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kamnitzer, J.: The crystal structure on the set of Mirković-Vilonen polytopes. Adv. Math 215(1), 66–93 (2007). MR 2354986 (2009a:17021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J 63(2), 465–516 (1991). MR 1115118 (93b:17045)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kashiwara, M.: The crystal base and Littelmann’s refined Demazure character formula. Duke Math. J 71(3), 839–858 (1993). MR 1240605 (95b:17019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kashiwara, M.: On crystal bases Representations of groups (Banff, AB). In: CMS conference proceedings, vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155–197. MR 1357199 (97a:17016) (1994)

  21. Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Affine crystals and vertex models Infinite analysis, Part A, B (Kyoto). Adv. Ser. Math. Phys 16, 449–484 (1991). World Sci. Publ., River Edge, NJ, 1992,. MR1187560 (94a:17008)

    MathSciNet  MATH  Google Scholar 

  22. Kerov, S.V., Kirillov, A.N., Reshetikhin, Y.N.: Combinatorics, the Bethe ansatz and representations of the symmetric group. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155(Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII), 50–64, 193 (1986). MR 869576 (88i:82021)

    MathSciNet  MATH  Google Scholar 

  23. Kang, S.-J., Misra, K.C.: Crystal bases and tensor product decompositions of U q (G 2)-modules. J. Algebra 163(3), 675–691 (1994). MR 1265857 (95f:17013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kashiwara, M., Misra, K.C., Okado, M., Yamada, D.: Perfect crystals for \(U_q(D_4^{(3)})\). J. Algebra 317(1), 392–423 (2007). MR 2360156 (2009b:17035)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165(2), 295–345 (1994). MR 1273277 (95c:17025)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kirillov, A.N., Reshetikhin, Y.N.: The Bethe ansatz and the combinatorics of Young tableaux. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155(Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII), 65–115, 194 (1986). MR 869577 (88i:82020)

    MathSciNet  MATH  Google Scholar 

  27. Kirillov, A.N., Schilling, A., Shimozono, M.: A bijection between Littlewood-Richardson tableaux and rigged configurations. Selecta Math. (N.S.) 8(1), 67–135 (2002). MR 1890195 2003a:05151

    Article  MathSciNet  MATH  Google Scholar 

  28. Kamnitzer, J., Tingley, P.: A definition of the crystal commutor using Kashiwara’s involution. J. Algebraic Combin 29(2), 261–268 (2009). MR 2475637 2010e:17011

    Article  MathSciNet  MATH  Google Scholar 

  29. Littelmann, P.: Cones, crystals, and patterns. Transform Groups 3(2), 145–179 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lee, K.-H., Salisbury, B.: A combinatorial description of the Gindikin-Karpelevich formula in type A. J. Combin. Theory Ser. A 119(5), 1081–1094 (2012). MR 2891384

    Article  MathSciNet  MATH  Google Scholar 

  31. Lee, K.-H., Salisbury, B.: Young tableaux, canonical bases, and the Gindikin-Karpelevich formula. J. Korean Math. Soc 51(2), 289–309 (2014). MR 3178585

    Article  MathSciNet  MATH  Google Scholar 

  32. Lusztig, G.: Introduction to quantum groups. MR 1227098 (94m:17016), vol. 110. Birkhäuser Boston, Inc., Boston (1993)

  33. Nakajima, H.: t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras. Represent. Theory 7, 259–274 (electronic) (2003). MR 1993360 (2004e:17013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Okado, M., Sano, N., KKR type bijection for the exceptional affine algebra \(E_6^{(1)}\) Algebraic groups and quantum groups. Contemp. Math. 565, 227–242 (2012). Amer. Math. Soc., Providence, RI, MR 2932429

  35. Okado, M., Schilling, A., Shimozono, M.: A crystal to rigged configuration bijection for nonexceptional affine algebras Algebraic combinatorics and quantum groups, 85–124 (2003). MR2035131 (2005b:17037)

  36. Okado, M., Schilling, A., Shimozono, M.: Virtual crystals and fermionic formulas of type \(D_{n+1}^{(2)}\), \(A_{2n}^{(2)}\), and \(C_n^{(1)}\). Represent Theory 7, 101–163 (electronic) (2003). MR1973369 (2004f:17023)

  37. Okado, M., Schilling, A., Shimozono, M.: Virtual crystals and Kleber’s algorithm. Comm. Math. Phys 238(1-2), 187–209 (2003). MR 1989674 (2004c:17034)

    Article  MathSciNet  MATH  Google Scholar 

  38. Okado, M., Sakamoto, R., Schilling, A.: Affine crystal structure on rigged configurations of type \(D_{n}^{(1)}\). J. Algebraic Combin 37(3), 571–599 (2013). MR 3035517

    Article  MathSciNet  MATH  Google Scholar 

  39. The Sage Developers, Sage Mathematics Software (Version 6.9) 2015, http://www.sagemath.org

  40. Sakamoto, R.: Rigged configurations and Kashiwara operators. SIGMA Symmetry Integrability Geom. Methods Appl 10 (2014). Paper 028, 88. MR3210607

  41. The Sage-Combinat community, Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics 2008, http://combinat.sagemath.org

  42. Schilling, A.: A bijection between, type \(D^{(1)_{n}}\) crystals and rigged configurations. J. Algebra 285(1), 292–334 (2005). MR 2119115 (2006i:17025)

    Article  MathSciNet  MATH  Google Scholar 

  43. Schilling, A.: Crystal structure on rigged configurations. Int. Math. Res. Not. (2006), Art. ID 97376 27. MR2211139 (2007i:17021)

  44. Scrimshaw, T.: A crystal to rigged configuration bijection and the filling map for type \(D_{4}^{(3)}\). J. Algebra 448C, 294–349 (2016)

  45. Schilling, A., Shimozono, M.: X = M for symmetric powers. J. Algebra 295(2), 562–610 (2006). MR 2194969 (2007a:17025)

    Article  MathSciNet  MATH  Google Scholar 

  46. Salisbury, B., Scrimshaw, T.: A rigged configuration model for \(B(\infty )\). J. Combin. Theory Ser. A 133, 29–57 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. Schilling, A., Scrimshaw, T.: Crystal structure on rigged configurations and the filling map. Electron. J. Combin 22(1) (2015). Research Paper 73, 56

  48. Tingley, P., Webster, B. Mirkovic-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras (2015). Preprint. arxiv: 1210.692

  49. Yamane, S.: Perfect crystals of \(U_q(G_2^{(1)})\). J. Algebra 210(2), 440–486 (1998). MR 1662347 (2000f:17024)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Author notes

  1. Travis Scrimshaw

    Present address: School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

Authors and Affiliations

  1. Department of Mathematics, Central Michigan University, Mt. Pleasant, MI, 48859, USA

    Ben Salisbury

  2. Department of Mathematics, University of California, Davis, CA, 95616, USA

    Travis Scrimshaw

Authors
  1. Ben Salisbury
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  2. Travis Scrimshaw
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Corresponding author

Correspondence to Ben Salisbury.

Additional information

Presented by Peter Littelmann.

T.S. was partially supported by NSF grant OCI-1147247.

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Salisbury, B., Scrimshaw, T. Connecting Marginally Large Tableaux and Rigged Configurations via Crystals. Algebr Represent Theor 19, 523–546 (2016). https://doi.org/10.1007/s10468-015-9587-y

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  • DOI: https://doi.org/10.1007/s10468-015-9587-y

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