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Hibi Algebras and Representation Theory

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Abstract

This paper gives a survey on the relation between Hibi algebras and representation theory. The notion of Hodge algebras or algebras with straightening laws has been proved to be very useful to describe the structure of many important algebras in classical invariant theory and representation theory (Bruns and Herzog 1993; De Concini et al. 1982; Eisenbud 1980; Gonciulea and Lakshmibai 2001; Seshadri 2007). In particular, a special type of such algebras introduced by Hibi (1987) provides a nice bridge between combinatorics and representation theory of classical groups. We will examine certain poset structures of Young tableaux and affine monoids, Hibi algebras in toric degenerations of flag varieties, and their relations to polynomial representations of the complex general linear group.

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References

  1. Berele, A.: Construction of \(\text {Sp}\)-modules by tableaux. Linear Multilinear Algebra 19(4), 299–307 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bruns, W., Herzog, J.: Cohen-Macaulay Rings Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  3. Conca, A., Herzog, J., Valla, G.: Sagbi bases with applications to blow-up algebras. J. Reine Angew. Math. 474, 113–138 (1996)

    MathSciNet  MATH  Google Scholar 

  4. De Concini, C.: Symplectic standard tableaux. Adv. Math. 34(1), 1–27 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. De Concini, C., Eisenbud, D., Procesi, C.: Hodge Algebras. Astérisque, 91. Société Mathématique de France, Paris 87 pp. (1982)

  6. Eisenbud, D.: Introduction to algebras with straightening laws. Ring theory and algebra, III (Proc. Third Conf., Univ. Oklahoma, Norman, Okla., 1979), Lecture Notes in Pure and Appl Math., vol. 55, pp 243–268. Dekker, New York (1980)

    Google Scholar 

  7. Fulton, W.: Young Tableaux. With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  8. Gelfand, I.M., Tsetlin, M.L.: Finite-dimensional representations of the group of unimodular matrices. Doklady Akad. Nauk SSSR (N.S.) 71, 825–828 (1950). English translation in Izrail M. Gelfand, Collected papers. Vol. II. Springer, Berlin (1988)

    MathSciNet  MATH  Google Scholar 

  9. Gonciulea, N., Lakshmibai, V.: Degenerations of flag and Schubert varieties to toric varieties. Transform. Groups 1(3), 215–248 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gonciulea, N., Lakshmibai, V.: Flag Varieties Hermann Éditeurs des Sciences et des Arts (2001)

  11. Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants Graduate Texts in Mathematics, vol. 255. Springer, Dordrecht (2009)

    Book  Google Scholar 

  12. Hibi, T.: Distributive lattices, affine semigroup rings and algebras with straightening laws. Commutative Algebra and Combinatorics (Kyoto, 1985), vol. 11, pp 93–109. Adv. Stud. Pure Math., Amsterdam (1987)

    Google Scholar 

  13. Hodge, W.V.D., Pedoe, D.: Methods of Algebraic Geometry. Vol. II. Reprint of the 1952 original. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  14. Howe, R.: Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. The Schur lectures (1992) (Tel Aviv), pp. 1–182, Israel Math. Conf Proc., vol. 8. Bar-Ilan Univ., Ramat Gan (1995)

    Google Scholar 

  15. Howe, R.: Weyl Chambers and standard monomial theory for poset lattice cones. Q. J. Pure Appl. Math. 1(1), 227–239 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Howe, R.: Pieri algebras and Hibi algebras in representation theory. Symmetry: Representation Theory and its Applications, pp. 353–384, Progr Math., vol. 257. Birkhäuser/Springer, New York (2014)

    Google Scholar 

  17. Howe, R., Lee, S.T.: Why should the Littlewood-Richardson rule be true. Bull. Am. Math. Soc. (N.S.) 49(2), 187–236 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Howe, R., Kim, S., Lee, S.T.: Double Pieri algebras and iterated Pieri algebras for the classical groups. Am. J. Math. 139(2), 347–401 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Howe, R., Kim, S., Lee, S.T.: Standard monomial theory for harmonics in classical invariant theory. Representation Theory, Number Theory and Invariant Theory, pp. 265–302. Progr Math., vol. 323. Birkhäuser/Springer, New York (2017)

    Google Scholar 

  20. Kim, S.: Standard monomial theory for flag algebras of \({{{GL}}}(n)\) and \({Sp}(2n)\). J. Algebra 320(2), 534–568 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kim, S.: The nullcone in the multi-vector representation of the symplectic group and related combinatorics. J. Combin. Theory Ser. A 117(8), 1231–1247 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kim, S.: Distributive lattices, affine semigroups, and branching rules of the classical groups. J. Combin. Theory Ser. A 119, 1132–1157 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kim, S.: A presentation of the double Pieri algebra. J. Pure Appl. Algebra 222 (2), 368–381 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kim, S., Lee, S.T.: Pieri algebras for the orthogonal and symplectic groups. Israel J. Math. 195(1), 215–245 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kim, S., Yacobi, O.: A basis for the symplectic group branching algebra. J. Algebraic Combin. 35(2), 269–290 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kim, S., Yoo, S.: Pieri and Littlewood-Richardson rules for two rows and cluster algebra structure. J. Algebraic Combin. 45(3), 887–909 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. King, R.C., El-Sharkaway, N.G.I.: Standard Young tableaux and weight multiplicities of the classical Lie groups. J. Phys. A 16(14), 3153–3177 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kogan, M., Miller, E.: Toric degeneration of Schubert varieties and Gelfand-Tsetlin polytopes. Adv. Math. 193(1), 1–17 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)

    Google Scholar 

  30. Molev, A.I.: Gelfand-Tsetlin bases for classical lie algebras. Handbook of Algebra 4, pp. 109–170, Handb Algebr., vol. 4. Elsevier/North-Holland, Amsterdam (2006)

    Google Scholar 

  31. Proctor, R.A.: Young tableaux, Gelfand patterns, and branching rules for classical groups. J. Algebra 164(2), 299–360 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  32. Seshadri, C.S.: Introduction to the Theory of Standard Monomials Texts and Readings in Mathematics, vol. 46. Hindustan Book Agency, New Delhi (2007)

    Google Scholar 

  33. Stanley, R.P.: Enumerative combinatorics. Vol. 1. 2nd edition. Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  34. Sturmfels, B.: Algorithms in invariant theory. Texts and Monographs in Symbolic Computation. Springer, Vienna (1993)

    Google Scholar 

  35. Wang, Y.: Sign Hibi cones and the anti-row iterated Pieri algebras for the general linear groups. J. Algebra 410, 355–392 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Parts of this article were presented at The Prospects for Commutative Algebra, Osaka, Japan, July 2017. We express our sincere gratitude to the organizers for the wonderful and stimulating conference.

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Authors and Affiliations

  1. Department of Mathematics, Korea University, Seoul, 02841, South Korea

    Sangjib Kim

  2. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Victor Protsak

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  1. Sangjib Kim
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Correspondence to Sangjib Kim.

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Kim, S., Protsak, V. Hibi Algebras and Representation Theory. Acta Math Vietnam 44, 307–323 (2019). https://doi.org/10.1007/s40306-018-0263-2

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