×

Bases for some reciprocity algebras. I. (English) Zbl 1135.22013

Let \(n,l,k\in \mathbb N\) be such that \(l_1+\ldots+l_k=l\) and \(l_j\leq n\). Consider the algebra \(P(\mathbb C^n\otimes \mathbb C^l)\) of polynomial functions on the space \(\mathbb C^n\otimes \mathbb C^l\) and the natural \(\text{GL}_n(\mathbb C)\times \text{GL}_l(\mathbb C)\). Further let \(U_m\) be the standard maximal unipotent subgroup of \(\text{GL}_m(\mathbb C)\). Then the algebra \(P(\mathbb C^n\otimes \mathbb C^l)^{U_n\times (U_{l_1}\times U_{l_k})}\) of \(U_n\times (U_{l_1}\times U_{l_k})\) invariants in \(P(\mathbb C^n\otimes \mathbb C^l)\) can be used to describe two different branching rules. The first determines the restriction of the \(k\)-fold tensor product of an irreducible \(\text{GL}_n(\mathbb C)\)-module to \(\text{GL}_n(\mathbb C)\). The second gives the decomposition of an irreducible \(\text{GL}_l(\mathbb C)\)-module under the action of \(\text{GL}_{l_1}(\mathbb C)\times \text{GL}_{l_k}(\mathbb C)\). In the notation in a recent set of papers by R. Howe, E. Tan and J. F. Willenbring [Adv. Math. 196, 531–564 (2005; Zbl 1072.22007); Trans. Am. Math. Soc. 357, 1601–1626 (2005; Zbl 1069.22006)], where this Howe duality approach to branching laws was introduced, \(P({\mathbb C}^n\otimes \mathbb C^l)^{U_n\times (U_{l_1}\times U_{l_k})}\) is called a reciprocity algebra. In this paper the authors construct a linear basis \(\{x_T : T\in A\}\) for this algebra. The elements of the basis are indexed by a set \(A\) of Littlewood-Richardson tableaux. An analogous program is carried out in this paper for the algebra \(P(\mathbb C^n\times \mathbb C^k)\oplus({\mathbb C^n}^*\otimes \mathbb C^l)\) with its natural \(\text{GL}_n(\mathbb C)\times \text{GL}_k(\mathbb C)\times \text{GL}_l(\mathbb C)\) action. More examples can be found in the authors’ follow-up papers [Adv. Math. 206, 145–210 (2006; Zbl 1106.22011), and Compos. Math. 142, No. 6, 1594–1614 (2006; Zbl 1130.22007)].

MSC:

22E46 Semisimple Lie groups and their representations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. M. Benkart, D. J. Britten, and F. W. Lemire, Stability in modules for classical Lie algebras — a constructive approach, Mem. Amer. Math. Soc. 85 (1990), no. 430, vi+165. · Zbl 0706.17003
[2] M. Brion and V. Lakshmibai, A geometric approach to standard monomial theory, Represent. Theory 7 (2003), 651 – 680. · Zbl 1053.14056
[3] A. D. Berenstein and A. V. Zelevinsky, Triple multiplicities for \?\?(\?+1) and the spectrum of the exterior algebra of the adjoint representation, J. Algebraic Combin. 1 (1992), no. 1, 7 – 22. · Zbl 0799.17005
[4] Philippe Caldero, Toric degenerations of Schubert varieties, Transform. Groups 7 (2002), no. 1, 51 – 60. · Zbl 1050.14040
[5] R. Chirivì, LS algebras and application to Schubert varieties, Transform. Groups 5 (2000), no. 3, 245 – 264. · Zbl 1019.14019
[6] David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, 2nd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997. An introduction to computational algebraic geometry and commutative algebra. · Zbl 0756.13017
[7] Corrado De Concini, David Eisenbud, and Claudio Procesi, Hodge algebras, Astérisque, vol. 91, Société Mathématique de France, Paris, 1982. With a French summary. · Zbl 0509.13026
[8] Thomas Enright, Roger Howe, and Nolan Wallach, A classification of unitary highest weight modules, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 97 – 143. · Zbl 0535.22012
[9] William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. · Zbl 0878.14034
[10] N. Gonciulea and V. Lakshmibai, Degenerations of flag and Schubert varieties to toric varieties, Transform. Groups 1 (1996), no. 3, 215 – 248. · Zbl 0909.14028
[11] Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. · Zbl 0901.22001
[12] W. V. D. Hodge, Some enumerative results in the theory of forms, Proc. Cambridge Philos. Soc. 39 (1943), 22 – 30. · Zbl 0060.04107
[13] Roger Howe, Reciprocity laws in the theory of dual pairs, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 159 – 175.
[14] Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539 – 570. , https://doi.org/10.1090/S0002-9947-1989-0986027-X Roger Howe, Erratum to: ”Remarks on classical invariant theory”, Trans. Amer. Math. Soc. 318 (1990), no. 2, 823. · Zbl 0674.15021
[15] Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1 – 182. · Zbl 0194.53802
[16] R. Howe, E-C. Tan and J. Willenbring, Reciprocity Algebras and Branching for Classical Symmetric Pairs. · Zbl 1176.22012
[17] Roger Howe, Eng-Chye Tan, and Jeb F. Willenbring, Stable branching rules for classical symmetric pairs, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1601 – 1626. · Zbl 1069.22006
[18] Roger E. Howe, Eng-Chye Tan, and Jeb F. Willenbring, A basis for the \?\?_{\?} tensor product algebra, Adv. Math. 196 (2005), no. 2, 531 – 564. · Zbl 1072.22007
[19] Mikhail Kogan and Ezra Miller, Toric degeneration of Schubert varieties and Gelfand-Tsetlin polytopes, Adv. Math. 193 (2005), no. 1, 1 – 17. · Zbl 1084.14049
[20] Stephen S. Kudla, Seesaw dual reductive pairs, Automorphic forms of several variables (Katata, 1983) Progr. Math., vol. 46, Birkhäuser Boston, Boston, MA, 1984, pp. 244 – 268. · Zbl 0549.10017
[21] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, 1 – 47. · Zbl 0375.22009
[22] V. Lakshmibai, Geometry of \?/\?. VI. Bases for fundamental representations of classical groups, J. Algebra 108 (1987), no. 2, 355 – 402. , https://doi.org/10.1016/0021-8693(87)90108-6 V. Lakshmibai, Geometry of \?/\?. VII. The symplectic group and the involution \?, J. Algebra 108 (1987), no. 2, 403 – 434. , https://doi.org/10.1016/0021-8693(87)90109-8 V. Lakshmibai, Geometry of \?/\?. VIII. The groups \?\?(2\?+1) and the involution \?, J. Algebra 108 (1987), no. 2, 435 – 471.
[23] V. Lakshmibai, Geometry of \?/\?. VI. Bases for fundamental representations of classical groups, J. Algebra 108 (1987), no. 2, 355 – 402. , https://doi.org/10.1016/0021-8693(87)90108-6 V. Lakshmibai, Geometry of \?/\?. VII. The symplectic group and the involution \?, J. Algebra 108 (1987), no. 2, 403 – 434. , https://doi.org/10.1016/0021-8693(87)90109-8 V. Lakshmibai, Geometry of \?/\?. VIII. The groups \?\?(2\?+1) and the involution \?, J. Algebra 108 (1987), no. 2, 435 – 471.
[24] V. Lakshmibai, Geometry of \?/\?. VI. Bases for fundamental representations of classical groups, J. Algebra 108 (1987), no. 2, 355 – 402. , https://doi.org/10.1016/0021-8693(87)90108-6 V. Lakshmibai, Geometry of \?/\?. VII. The symplectic group and the involution \?, J. Algebra 108 (1987), no. 2, 403 – 434. , https://doi.org/10.1016/0021-8693(87)90109-8 V. Lakshmibai, Geometry of \?/\?. VIII. The groups \?\?(2\?+1) and the involution \?, J. Algebra 108 (1987), no. 2, 435 – 471.
[25] V. Lakshmibai, Standard monomial theory for \Hat \?\?_{\?}, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 197 – 217. · Zbl 1022.90014
[26] Venkatramani Lakshmibai, Peter Littelmann, and Peter Magyar, Standard monomial theory and applications, Representation theories and algebraic geometry (Montreal, PQ, 1997) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 514, Kluwer Acad. Publ., Dordrecht, 1998, pp. 319 – 364. Notes by Rupert W. T. Yu. · Zbl 1013.17014
[27] C. S. Seshadri, Geometry of \?/\?. I. Theory of standard monomials for minuscule representations, C. P. Ramanujam — a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 207 – 239. V. Lakshmibai and C. S. Seshadri, Geometry of \?/\?. II. The work of de Concini and Procesi and the basic conjectures, Proc. Indian Acad. Sci. Sect. A 87 (1978), no. 2, 1 – 54. V. Lakshmibai, C. Musili, and C. S. Seshadri, Geometry of \?/\?. III. Standard monomial theory for a quasi-minuscule \?, Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), no. 3, 93 – 177. V. Lakshmibai, C. Musili, and C. S. Seshadri, Geometry of \?/\?. IV. Standard monomial theory for classical types, Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), no. 4, 279 – 362. · Zbl 0447.14011
[28] C. S. Seshadri, Geometry of \?/\?. I. Theory of standard monomials for minuscule representations, C. P. Ramanujam — a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 207 – 239. V. Lakshmibai and C. S. Seshadri, Geometry of \?/\?. II. The work of de Concini and Procesi and the basic conjectures, Proc. Indian Acad. Sci. Sect. A 87 (1978), no. 2, 1 – 54. V. Lakshmibai, C. Musili, and C. S. Seshadri, Geometry of \?/\?. III. Standard monomial theory for a quasi-minuscule \?, Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), no. 3, 93 – 177. V. Lakshmibai, C. Musili, and C. S. Seshadri, Geometry of \?/\?. IV. Standard monomial theory for classical types, Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), no. 4, 279 – 362. · Zbl 0447.14011
[29] V. Lakshmibai and C. S. Seshadri, Standard monomial theory, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) Manoj Prakashan, Madras, 1991, pp. 279 – 322. · Zbl 0785.14028
[30] I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. · Zbl 0487.20007
[31] Lorenzo Robbiano and Moss Sweedler, Subalgebra bases, Commutative algebra (Salvador, 1988) Lecture Notes in Math., vol. 1430, Springer, Berlin, 1990, pp. 61 – 87. · Zbl 0725.13013
[32] Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. · Zbl 0856.13020
[33] Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. · Zbl 1024.20501
[34] Компактные группы Ли и их представления., Издат. ”Наука”, Мосцощ, 1970 (Руссиан). Д. П. Žелобенко, Цомпацт Лие гроупс анд тхеир репресентатионс, Америцан Матхематицал Социеты, Провиденце, Р.И., 1973. Транслатед фром тхе Руссиан бы Исраел Програм фор Сциентифиц Транслатионс; Транслатионс оф Матхематицал Монограпхс, Вол. 40.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.