Standard monomial theory

{{harvs|txt|first=Alfred|last=Young|year=1928|authorlink=Alfred Young (mathematician)}} introduced monomials associated to standard Young tableaux.

{{harvs|txt|last=Hodge|authorlink=W. V. D. Hodge|year=1943}} (see also {{harv|Hodge|Pedoe|1994|loc=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. {{harvs|txt|last=Seshadri|authorlink=C. S. Seshadri|year=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. {{harvs|txt|last=Littelmann|authorlink=Peter Littelmann|year=1998}} proved their conjectures using the Littelmann path model, in particular giving a uniform description of standard monomials for all reductive groups.

{{harvtxt|Lakshmibai|2003}} and {{harvtxt|Musili|2003}} and {{harvtxt|Seshadri|2012}} give detailed descriptions of the early development of standard monomial theory.

• 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.

{{reflist}}

• {{Citation | last1=Hodge | first1=W. V. D. | author1-link=W. V. D. Hodge | title=Some enumerative results in the theory of forms | doi=10.1017/S0305004100017631 |mr=0007739 | year=1943 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | volume=39 | issue=1 | pages=22–30| bibcode=1943PCPS...39...22H }}
• {{Citation | last1=Hodge | first1=W. V. D. | author1-link=W. V. D. Hodge | last2=Pedoe | first2=Daniel | author2-link=Daniel Pedoe | title=Methods of Algebraic Geometry: Volume 2 Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties. | orig-year=1952 | url=https://books.google.com/books?id=bJwbn3RSWhwC | publisher=Cambridge University Press | series=Cambridge Mathematical Library | isbn=978-0-521-46901-2 |mr=0048065 | year=1994}}
• {{Citation | last1=Lakshmibai | first1=V. | last2=Musili | first2=C. | last3=Seshadri | first3=C. S. | title=Geometry of G/P | doi=10.1090/S0273-0979-1979-14631-7 |mr=520081 | year=1979 | journal=Bulletin of the American Mathematical Society |series=New Series | issn=0002-9904 | volume=1 | issue=2 | pages=432–435| doi-access=free }}
• {{Citation | last1=Lakshmibai | first1=Venkatramani | last2=Raghavan | first2=Komaranapuram N. | title=Standard monomial theory | publisher=Springer-Verlag | location=Berlin, New York | series=Encyclopaedia of Mathematical Sciences | isbn=978-3-540-76756-5 | doi=10.1007/978-3-540-76757-2 |mr=2388163 | year=2008 | volume=137}}
• {{Citation | last1=Lakshmibai | first1=V. | last2=Seshadri | first2=C. S. | editor1-last=Ramanan | editor1-first=S. | editor2-last=Musili | editor2-first=C. | editor3-last=Kumar | editor3-first=N. Mohan | title=Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) | chapter-url=https://books.google.com/books?id=r\_3uAAAAMAAJ | publisher=Manoj Prakashan | location=Madras |mr=1131317 | year=1991 | chapter=Standard monomial theory | pages=279–322}}
• {{Citation | last1=Lakshmibai | first1=V. | title=A tribute to C. S. Seshadri (Chennai, 2002) | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Trends Math. | isbn=978-3-7643-0444-7 |mr=2017589 | year=2003 | chapter=The development of standard monomial theory. II | chapter-url=https://books.google.com/books?id=sjGmWWicE10C&pg=PA283 | pages=283–309}}
• {{Citation | last1=Littelmann | first1=Peter | editor1-last=Carter | editor1-first=Roger W. | editor1-link=Roger Carter (mathematician) | editor2-last=Saxl | editor2-first=Jan | title=Algebraic groups and their representations (Cambridge, 1997) | url=https://books.google.com/books?id=5rJ1t0aNuygC&pg=PA175 | publisher=Kluwer Academic Publishers Group | location=Dordrecht | series= NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences |mr=1670770 | year=1998 | volume=517 | chapter=The path model, the quantum Frobenius map and standard monomial theory | chapter-url=http://www.mi.uni-koeln.de/~littelma/cambridge.pdf | pages=175–212| isbn=9780792352921 }}
• {{Citation | last1=Littelmann | first1=Peter | title=Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras | doi=10.1090/S0894-0347-98-00268-9 |mr=1603862 | year=1998 | journal=Journal of the American Mathematical Society | issn=0894-0347 | volume=11 | issue=3 | pages=551–567| doi-access=free }}
• {{Citation | last1=Musili | first1=C. | title=A tribute to C. S. Seshadri (Chennai, 2002) | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Trends Math. | isbn=978-3-7643-0444-7 |mr=2017594 | year=2003 | chapter=The development of standard monomial theory. I | chapter-url=https://books.google.com/books?id=sjGmWWicE10C&pg=PA385 | pages=385–420}}
• {{Citation | last1=Seshadri | first1=C. S. | editor1-last=Ramanathan | editor1-first=K. G. | title=C. P. Ramanujam---a tribute | publisher=Springer-Verlag | location=Berlin, New York | series= Tata Institute of Fundamental Research Studies in Mathematics | isbn=978-3-540-08770-0 |mr=541023 | year=1978 | volume=8 | chapter=Geometry of G/P. I. Theory of standard monomials for minuscule representations | pages=207–239}}
• {{Citation | last1=Seshadri | first1=C. S. | title=Introduction to the theory of standard monomials | url=http://www.ams.org/bookstore-getitem/item=HIN-34 | publisher=Hindustan Book Agency | location=New Delhi | series=Texts and Readings in Mathematics | isbn=9788185931784 |mr=2347272 | year=2007 | volume=46}}
• {{Citation | last1=Seshadri | first1=C. S. | title=Collected papers of C. S. Seshadri. Volume 2. Schubert geometry and representation theory. | chapter-url=http://www.ams.org/bookstore-getitem/item=HIN-50 | publisher=Hindustan Book Agency | location=New Delhi | isbn=9789380250175 |mr=2905898 | year=2012|chapter=Standard monomial theory — a historical account|pages=3–50}}
• {{Citation | last1=Young | first1=Alfred | author1-link=Alfred Young (mathematician) | title=On Quantitative Substitutional Analysis | doi=10.1112/plms/s2-28.1.255 | year=1928 | journal= Proc. London Math. Soc. | volume=28 | issue=1 | pages=255–292| url=https://zenodo.org/record/1447746 }}

Category:Algebraic geometry

Category:Invariant theory