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Standard monomial theory

May 09, 2024

In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic group by giving an explicit basis of elements called standard monomials. Many of the results have been extended to Kac–Moody algebras and their groups.

There are monographs on standard monomial theory by Lakshmibai & Raghavan (2008) and Seshadri (2007) and survey articles by V. Lakshmibai, C. Musili, and C. S. Seshadri (1979) and V. Lakshmibai and C. S. Seshadri (1991).

One of important open problems is to give a completely geometric construction of the theory.[1]

History edit

Alfred Young (1928) introduced monomials associated to standard Young tableaux. Hodge (1943) (see also (Hodge & Pedoe 1994, p.378)) used Young's monomials, which he called standard power products, named after standard tableaux, to give a basis for the homogeneous coordinate rings of complex Grassmannians. Seshadri (1978) initiated a program, called standard monomial theory, to extend Hodge's work to varieties G/P, for P any parabolic subgroup of any reductive algebraic group in any characteristic, by giving explicit bases using standard monomials for sections of line bundles over these varieties. The case of Grassmannians studied by Hodge corresponds to the case when G is a special linear group in characteristic 0 and P is a maximal parabolic subgroup. Seshadri was soon joined in this effort by V. Lakshmibai and Chitikila Musili. They worked out standard monomial theory first for minuscule representations of G and then for groups G of classical type, and formulated several conjectures describing it for more general cases. Littelmann (1998) proved their conjectures using the Littelmann path model, in particular giving a uniform description of standard monomials for all reductive groups.

Lakshmibai (2003) and Musili (2003) and Seshadri (2012) give detailed descriptions of the early development of standard monomial theory.

Applications edit

  • Since the sections of line bundles over generalized flag varieties tend to form irreducible representations of the corresponding algebraic groups, having an explicit basis of standard monomials allows one to give character formulas for these representations. Similarly one gets character formulas for Demazure modules. The explicit bases given by standard monomial theory are closely related to crystal bases and Littelmann path models of representations.
  • Standard monomial theory allows one to describe the singularities of Schubert varieties, and in particular sometimes proves that Schubert varieties are normal or Cohen–Macaulay.
  • Standard monomial theory can be used to prove Demazure's conjecture.
  • Standard monomial theory proves the Kempf vanishing theorem and other vanishing theorems for the higher cohomology of effective line bundles over Schubert varieties.
  • Standard monomial theory gives explicit bases for some rings of invariants in invariant theory.
  • Standard monomial theory gives generalizations of the Littlewood–Richardson rule about decompositions of tensor products of representations to all reductive algebraic groups.
  • Standard monomial theory can be used to prove the existence of good filtrations on some representations of reductive algebraic groups in positive characteristic.

Notes edit

  1. ^ M. Brion and V. Lakshmibai : A geometric approach to standard monomial theory, Represent. Theory 7 (2003), 651–680.

References edit

  • Hodge, W. V. D. (1943), "Some enumerative results in the theory of forms", Proc. Cambridge Philos. Soc., 39 (1): 22–30, Bibcode:1943PCPS...39...22H, doi:10.1017/S0305004100017631, MR 0007739
  • Hodge, W. V. D.; Pedoe, Daniel (1994) [1952], Methods of Algebraic Geometry: Volume 2 Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties., Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-46901-2, MR 0048065
  • Lakshmibai, V.; Musili, C.; Seshadri, C. S. (1979), "Geometry of G/P", Bulletin of the American Mathematical Society, New Series, 1 (2): 432–435, doi:10.1090/S0273-0979-1979-14631-7, ISSN 0002-9904, MR 0520081
  • Lakshmibai, Venkatramani; Raghavan, Komaranapuram N. (2008), Standard monomial theory, Encyclopaedia of Mathematical Sciences, vol. 137, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-76757-2, ISBN 978-3-540-76756-5, MR 2388163
  • Lakshmibai, V.; Seshadri, C. S. (1991), "Standard monomial theory", in Ramanan, S.; Musili, C.; Kumar, N. Mohan (eds.), Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Madras: Manoj Prakashan, pp. 279–322, MR 1131317
  • Lakshmibai, V. (2003), "The development of standard monomial theory. II", A tribute to C. S. Seshadri (Chennai, 2002), Trends Math., Basel, Boston, Berlin: Birkhäuser, pp. 283–309, ISBN 978-3-7643-0444-7, MR 2017589
  • Littelmann, Peter (1998), "The path model, the quantum Frobenius map and standard monomial theory" (PDF), in Carter, Roger W.; Saxl, Jan (eds.), Algebraic groups and their representations (Cambridge, 1997), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 517, Dordrecht: Kluwer Academic Publishers Group, pp. 175–212, ISBN 9780792352921, MR 1670770
  • Littelmann, Peter (1998), "Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras", Journal of the American Mathematical Society, 11 (3): 551–567, doi:10.1090/S0894-0347-98-00268-9, ISSN 0894-0347, MR 1603862
  • Musili, C. (2003), "The development of standard monomial theory. I", A tribute to C. S. Seshadri (Chennai, 2002), Trends Math., Basel, Boston, Berlin: Birkhäuser, pp. 385–420, ISBN 978-3-7643-0444-7, MR 2017594
  • Seshadri, C. S. (1978), "Geometry of G/P. I. Theory of standard monomials for minuscule representations", in Ramanathan, K. G. (ed.), C. P. Ramanujam---a tribute, Tata Institute of Fundamental Research Studies in Mathematics, vol. 8, Berlin, New York: Springer-Verlag, pp. 207–239, ISBN 978-3-540-08770-0, MR 0541023
  • Seshadri, C. S. (2007), Introduction to the theory of standard monomials, Texts and Readings in Mathematics, vol. 46, New Delhi: Hindustan Book Agency, ISBN 9788185931784, MR 2347272
  • Seshadri, C. S. (2012), "Standard monomial theory — a historical account", Collected papers of C. S. Seshadri. Volume 2. Schubert geometry and representation theory., New Delhi: Hindustan Book Agency, pp. 3–50, ISBN 9789380250175, MR 2905898
  • Young, Alfred (1928), "On Quantitative Substitutional Analysis", Proc. London Math. Soc., 28 (1): 255–292, doi:10.1112/plms/s2-28.1.255

May 09, 2024
standard, monomial, theory, algebraic, geometry, standard, monomial, theory, describes, sections, line, bundle, over, generalized, flag, variety, schubert, variety, reductive, algebraic, group, giving, explicit, basis, elements, called, standard, monomials, ma. In algebraic geometry standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic group by giving an explicit basis of elements called standard monomials Many of the results have been extended to Kac Moody algebras and their groups There are monographs on standard monomial theory by Lakshmibai amp Raghavan 2008 and Seshadri 2007 and survey articles by V Lakshmibai C Musili and C S Seshadri 1979 and V Lakshmibai and C S Seshadri 1991 One of important open problems is to give a completely geometric construction of the theory 1 Contents 1 History 2 Applications 3 Notes 4 ReferencesHistory editAlfred Young 1928 introduced monomials associated to standard Young tableaux Hodge 1943 see also Hodge amp Pedoe 1994 p 378 used Young s monomials which he called standard power products named after standard tableaux to give a basis for the homogeneous coordinate rings of complex Grassmannians Seshadri 1978 initiated a program called standard monomial theory to extend Hodge s work to varieties G P for P any parabolic subgroup of any reductive algebraic group in any characteristic by giving explicit bases using standard monomials for sections of line bundles over these varieties The case of Grassmannians studied by Hodge corresponds to the case when G is a special linear group in characteristic 0 and P is a maximal parabolic subgroup Seshadri was soon joined in this effort by V Lakshmibai and Chitikila Musili They worked out standard monomial theory first for minuscule representations of G and then for groups G of classical type and formulated several conjectures describing it for more general cases Littelmann 1998 proved their conjectures using the Littelmann path model in particular giving a uniform description of standard monomials for all reductive groups Lakshmibai 2003 and Musili 2003 and Seshadri 2012 give detailed descriptions of the early development of standard monomial theory Applications editSince the sections of line bundles over generalized flag varieties tend to form irreducible representations of the corresponding algebraic groups having an explicit basis of standard monomials allows one to give character formulas for these representations Similarly one gets character formulas for Demazure modules The explicit bases given by standard monomial theory are closely related to crystal bases and Littelmann path models of representations Standard monomial theory allows one to describe the singularities of Schubert varieties and in particular sometimes proves that Schubert varieties are normal or Cohen Macaulay Standard monomial theory can be used to prove Demazure s conjecture Standard monomial theory proves the Kempf vanishing theorem and other vanishing theorems for the higher cohomology of effective line bundles over Schubert varieties Standard monomial theory gives explicit bases for some rings of invariants in invariant theory Standard monomial theory gives generalizations of the Littlewood Richardson rule about decompositions of tensor products of representations to all reductive algebraic groups Standard monomial theory can be used to prove the existence of good filtrations on some representations of reductive algebraic groups in positive characteristic Notes edit M Brion and V Lakshmibai A geometric approach to standard monomial theory Represent Theory 7 2003 651 680 References editHodge W V D 1943 Some enumerative results in the theory of forms Proc Cambridge Philos Soc 39 1 22 30 Bibcode 1943PCPS 39 22H doi 10 1017 S0305004100017631 MR 0007739 Hodge W V D Pedoe Daniel 1994 1952 Methods of Algebraic Geometry Volume 2 Book III General theory of algebraic varieties in projective space Book IV Quadrics and Grassmann varieties Cambridge Mathematical Library Cambridge University Press ISBN 978 0 521 46901 2 MR 0048065 Lakshmibai V Musili C Seshadri C S 1979 Geometry of G P Bulletin of the American Mathematical Society New Series 1 2 432 435 doi 10 1090 S0273 0979 1979 14631 7 ISSN 0002 9904 MR 0520081 Lakshmibai Venkatramani Raghavan Komaranapuram N 2008 Standard monomial theory Encyclopaedia of Mathematical Sciences vol 137 Berlin New York Springer Verlag doi 10 1007 978 3 540 76757 2 ISBN 978 3 540 76756 5 MR 2388163 Lakshmibai V Seshadri C S 1991 Standard monomial theory in Ramanan S Musili C Kumar N Mohan eds Proceedings of the Hyderabad Conference on Algebraic Groups Hyderabad 1989 Madras Manoj Prakashan pp 279 322 MR 1131317 Lakshmibai V 2003 The development of standard monomial theory II A tribute to C S Seshadri Chennai 2002 Trends Math Basel Boston Berlin Birkhauser pp 283 309 ISBN 978 3 7643 0444 7 MR 2017589 Littelmann Peter 1998 The path model the quantum Frobenius map and standard monomial theory PDF in Carter Roger W Saxl Jan eds Algebraic groups and their representations Cambridge 1997 NATO Advanced Science Institutes Series C Mathematical and Physical Sciences vol 517 Dordrecht Kluwer Academic Publishers Group pp 175 212 ISBN 9780792352921 MR 1670770 Littelmann Peter 1998 Contracting modules and standard monomial theory for symmetrizable Kac Moody algebras Journal of the American Mathematical Society 11 3 551 567 doi 10 1090 S0894 0347 98 00268 9 ISSN 0894 0347 MR 1603862 Musili C 2003 The development of standard monomial theory I A tribute to C S Seshadri Chennai 2002 Trends Math Basel Boston Berlin Birkhauser pp 385 420 ISBN 978 3 7643 0444 7 MR 2017594 Seshadri C S 1978 Geometry of G P I Theory of standard monomials for minuscule representations in Ramanathan K G ed C P Ramanujam a tribute Tata Institute of Fundamental Research Studies in Mathematics vol 8 Berlin New York Springer Verlag pp 207 239 ISBN 978 3 540 08770 0 MR 0541023 Seshadri C S 2007 Introduction to the theory of standard monomials Texts and Readings in Mathematics vol 46 New Delhi Hindustan Book Agency ISBN 9788185931784 MR 2347272 Seshadri C S 2012 Standard monomial theory a historical account Collected papers of C S Seshadri Volume 2 Schubert geometry and representation theory New Delhi Hindustan Book Agency pp 3 50 ISBN 9789380250175 MR 2905898 Young Alfred 1928 On Quantitative Substitutional Analysis Proc London Math Soc 28 1 255 292 doi 10 1112 plms s2 28 1 255 Retrieved from https en wikipedia org w index php title Standard monomial theory amp oldid 1219680428, wikipedia, wiki, book, books, library,

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