NCMW - Combinatorial models in Representation Theory (2019)

Speakers and Syllabus


  • Module A (B Ravinder, CMI) Young tableaux and the representation theory of the Lie algebra sln C: Review of some aspects of the general theory of representations of complex semisimple Lie algebras, the special case of sln C, highest weights and partitions, irreducible characters in terms of semistandard Young tableaux, Gelfand-Tsetlin patterns, Schur functions, the dimension formula, decomposing tensor products, the Littlewood- Richardson rule, Demazure modules.
  • Module B (Amritanshu Prasad, IMSc) The plactic monoid: Words, Knuth equivalence, plactic and coplactic classes, crystal operators on words, the Robinson-Schensted correspondence, Schur-Weyl duality.
  • Module C (KN Raghavan and S Viswanath, IMSc) The Littelmann path model: Paths, root operators, Lakshmibai-Seshadri paths, the character formula, tensor product decomposition in terms of paths, the Weyl group action, the crystal isomorphism theorem.
  • Module D (R Venkatesh, IISc) Affine Lie algebras and their representations: Affine Lie algebras as central extension of loop algebras, highest weight representations, Weyl-Kac character formula, character of the basic representation.
  • Module E (S Viswanath, IMSc) Affine crystals: An overview of various combinatorial models of crystals for the affine Lie algebra sln , such as partitions, LS paths, abacus, semi-infinite paths etc.
  • Module F (KN Raghavan, IMSc) Constructing nice bases for representations from path data: Using paths or other crystal models to construct bases with nice properties, such as compatibility with Demazure modules. Standard monomial theory, geometric applications.

Speakers

  1. B Ravinder (BR), CMI
  2. Amritanshu Prasad (AP), IMSc
  3. K N Raghavan (KNR), IMSc
  4. S Viswanath (SV), IMSc
  5. R Venkatesh (RV), IISc

 


Time Table

Tea Breaks: 11.00 - 11.30 and 3.30 - 4.00
Lunch Break: 1.00 - 2.00
Snacks: 5.00 - 5.30

  9.30 - 11.00
Lecture 1
11.30-1.00
Lecture 2
2.00-3.30
Lecture 3
4.00-5.00
Tutorial
Nov 04 BR AP KNR BR, AP, KNR
Nov 05 BR AP KNR BR, AP, KNR
Nov 06 BR AP KNR BR, AP, KNR
Nov 07 BR AP SV BR, AP, SV
Nov 08 BR AP SV BR, AP, SV
Nov 09 BR AP SV BR, AP, SV
Nov 11 RV SV KNR RV, SV, KNR
Nov 12 RV SV KNR RV, SV, KNR
Nov 13 RV SV KNR RV, SV, KNR
Nov 14 RV SV KNR RV, SV, KNR
Nov 15 RV SV KNR RV, SV, KNR
Nov 16 RV SV KNR RV, SV, KNR
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