mathematics

Standard monomial theory

Standard monomial theory is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Standard monomial theory rather than just read about it. In short: 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.

Key takeaways

Standard monomial theory belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Standard monomial theory to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Standard monomial theory from memory before moving on to harder problems.

Reference excerpt

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.

History 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 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

References Hodge, W. V. D. (1943), "Some enumerative results in the theory of forms", Mathematical Proceedings of the Cambridge Philosophical Society, 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

Worked examples

Example 1 — a first encounter with Standard monomial theory

Start with the simplest possible case. Write down what Standard monomial theory claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Standard monomial theory before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Standard monomial theory ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Standard monomial theory

In research: Standard monomial theory appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Standard monomial theory in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Standard monomial theory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, Invariant theory, so understanding it makes those chapters shorter.
In everyday life: Look for Standard monomial theory outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Standard monomial theory in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Standard monomial theory means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Standard monomial theory out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Standard monomial theory in simple terms?

Why does Standard monomial theory matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Standard monomial theory?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Standard monomial theory.