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Standard monomial theory
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<h2 id="history">History</h2> <p><a href="/facts/Alfred_Young_(mathematician)/L3R6cM6D">Alfred Young</a> (1928) introduced monomials associated to standard <a href="/facts/Young_tableaux/e5lI4bck">Young tableaux</a>. <a href="/facts/W._V._D._Hodge/5XsdVugb">Hodge</a> (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 <a href="/facts/Grassmannian/QcmgLRq8">Grassmannians</a>. <a href="/facts/C._S._Seshadri/YIrFP80W">Seshadri</a> (1978) initiated a program, called standard monomial theory, to extend Hodge's work to varieties <i>G</i>/<i>P</i>, for <i>P</i> any <a href="/facts/Borel_subgroup/F80yLMqA">parabolic subgroup</a> of any <a href="/facts/Reductive_algebraic_group/z1zyJvUh">reductive algebraic group</a> 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 <i>G</i> is a special linear group in characteristic 0 and <i>P</i> is a maximal parabolic subgroup. Seshadri was soon joined in this effort by V. Lakshmibai and <a href="/facts/Chitikila_Musili/mBrCBbJE">Chitikila Musili</a>. They worked out standard monomial theory first for <a href="/facts/Minuscule_representation/yacMsPBh">minuscule representations</a> of <i>G</i> and then for groups <i>G</i> of classical type, and formulated several conjectures describing it for more general cases. <a href="/facts/Peter_Littelmann/2dXHafXh">Littelmann</a> (1998) proved their conjectures using the <a href="/facts/Littelmann_path_model/DFuwMww1">Littelmann path model</a>, in particular giving a uniform description of standard monomials for all reductive groups. </p><p>Lakshmibai (2003) and Musili (2003) and Seshadri (2012) give detailed descriptions of the early development of standard monomial theory. </p> <h2 id="applications">Applications</h2> <ul><li>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 <a href="/facts/Demazure_module/1guQIDU8">Demazure modules</a>. The explicit bases given by standard monomial theory are closely related to <a href="/facts/Crystal_base/sb1qXgwa">crystal bases</a> and <a href="/facts/Littelmann_path_model/DFuwMww1">Littelmann path models</a> of representations.</li> <li>Standard monomial theory allows one to describe the singularities of Schubert varieties, and in particular sometimes proves that Schubert varieties are normal or <a href="/facts/Cohen%25E2%2580%2593Macaulay/4G8wymtC">Cohen–Macaulay</a>.</li> <li>Standard monomial theory can be used to prove <a href="/facts/Demazure%2527s_conjecture/Jl1Xho3m">Demazure's conjecture</a>.</li> <li>Standard monomial theory proves the <a href="/facts/Kempf_vanishing_theorem/1pxNwMEB">Kempf vanishing theorem</a> and other vanishing theorems for the higher cohomology of effective line bundles over Schubert varieties.</li> <li>Standard monomial theory gives explicit bases for some rings of invariants in <a href="/facts/Invariant_theory/5fIUh9h3">invariant theory</a>.</li> <li>Standard monomial theory gives generalizations of the <a href="/facts/Littlewood%25E2%2580%2593Richardson_rule/8NSx5K36">Littlewood–Richardson rule</a> about decompositions of tensor products of representations to all reductive algebraic groups.</li> <li>Standard monomial theory can be used to prove the existence of <a href="/facts/Good_filtration/9U8knpKR">good filtrations</a> on some representations of reductive algebraic groups in positive characteristic.</li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/W._V._D._Hodge/5XsdVugb">Hodge, W. V. D.</a> (1943), "Some enumerative results in the theory of forms", <i><a href="/facts/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society/KUUzcS7S">Mathematical Proceedings of the Cambridge Philosophical Society</a></i>, 39 (1): 22–30, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1943PCPS...39...22H">1943PCPS...39...22H</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FS0305004100017631">10.1017/S0305004100017631</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0007739">0007739</a></li> <li><a href="/facts/W._V._D._Hodge/5XsdVugb">Hodge, W. V. D.</a>; <a href="/facts/Daniel_Pedoe/uVGSaTQU">Pedoe, Daniel</a> (1994) [1952], <a href="https://books.google.com/books?id=bJwbn3RSWhwC"><i>Methods of Algebraic Geometry: Volume 2 Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties.</i></a>, Cambridge Mathematical Library, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-46901-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0048065">0048065</a></li> <li>Lakshmibai, V.; Musili, C.; Seshadri, C. S. (1979), "Geometry of G/P", <i>Bulletin of the American Mathematical Society</i>, New Series, 1 (2): 432–435, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0273-0979-1979-14631-7">10.1090/S0273-0979-1979-14631-7</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9904">0002-9904</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0520081">0520081</a></li> <li>Lakshmibai, Venkatramani; Raghavan, Komaranapuram N. (2008), <i>Standard monomial theory</i>, Encyclopaedia of Mathematical Sciences, vol. 137, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-540-76757-2">10.1007/978-3-540-76757-2</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-76756-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2388163">2388163</a></li> <li>Lakshmibai, V.; Seshadri, C. S. (1991), <a href="https://books.google.com/books?id=r\_3uAAAAMAAJ">"Standard monomial theory"</a>, in Ramanan, S.; Musili, C.; Kumar, N. Mohan (eds.), <i>Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989)</i>, Madras: Manoj Prakashan, pp. 279–322, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1131317">1131317</a></li> <li>Lakshmibai, V. (2003), <a href="https://books.google.com/books?id=sjGmWWicE10C&pg=PA283">"The development of standard monomial theory. II"</a>, <i>A tribute to C. S. Seshadri (Chennai, 2002)</i>, Trends Math., Basel, Boston, Berlin: Birkhäuser, pp. 283–309, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-7643-0444-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2017589">2017589</a></li> <li>Littelmann, Peter (1998), <a href="http://www.mi.uni-koeln.de/~littelma/cambridge.pdf">"The path model, the quantum Frobenius map and standard monomial theory"</a> (PDF), in <a href="/facts/Roger_Carter_(mathematician)/zuMTPjG0">Carter, Roger W.</a>; Saxl, Jan (eds.), <a href="https://books.google.com/books?id=5rJ1t0aNuygC&pg=PA175"><i>Algebraic groups and their representations (Cambridge, 1997)</i></a>, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 517, Dordrecht: Kluwer Academic Publishers Group, pp. 175–212, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780792352921, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1670770">1670770</a></li> <li>Littelmann, Peter (1998), "Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras", <i><a href="/facts/Journal_of_the_American_Mathematical_Society/qR98drCm">Journal of the American Mathematical Society</a></i>, 11 (3): 551–567, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0894-0347-98-00268-9">10.1090/S0894-0347-98-00268-9</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0894-0347">0894-0347</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1603862">1603862</a></li> <li>Musili, C. (2003), <a href="https://books.google.com/books?id=sjGmWWicE10C&pg=PA385">"The development of standard monomial theory. I"</a>, <i>A tribute to C. S. Seshadri (Chennai, 2002)</i>, Trends Math., Basel, Boston, Berlin: Birkhäuser, pp. 385–420, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-7643-0444-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2017594">2017594</a></li> <li>Seshadri, C. S. (1978), "Geometry of G/P. I. Theory of standard monomials for minuscule representations", in Ramanathan, K. G. (ed.), <i>C. P. Ramanujam---a tribute</i>, Tata Institute of Fundamental Research Studies in Mathematics, vol. 8, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, pp. 207–239, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-08770-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0541023">0541023</a></li> <li>Seshadri, C. S. (2007), <a href="http://www.ams.org/bookstore-getitem/item=HIN-34"><i>Introduction to the theory of standard monomials</i></a>, Texts and Readings in Mathematics, vol. 46, New Delhi: Hindustan Book Agency, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9788185931784, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2347272">2347272</a></li> <li>Seshadri, C. S. (2012), <a href="http://www.ams.org/bookstore-getitem/item=HIN-50">"Standard monomial theory — a historical account"</a>, <i>Collected papers of C. S. Seshadri. Volume 2. Schubert geometry and representation theory.</i>, New Delhi: Hindustan Book Agency, pp. 3–50, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9789380250175, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2905898">2905898</a></li> <li><a href="/facts/Alfred_Young_(mathematician)/L3R6cM6D">Young, Alfred</a> (1928), <a href="https://zenodo.org/record/1447746">"On Quantitative Substitutional Analysis"</a>, <i>Proc. London Math. Soc.</i>, 28 (1): 255–292, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fplms%2Fs2-28.1.255">10.1112/plms/s2-28.1.255</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>M. Brion and V. Lakshmibai : A geometric approach to standard monomial theory, Represent. Theory 7 (2003), 651–680. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>