From Littlewood-Richardson Coefficients to Cluster Algebras in Three Lectures

Lecture I presents a unified expression from [4] for generalized Littlewood- Richardson coefficients (= tensor product multiplicities) for any complex semisimple Lie algebra. Lecture II outlines a proof of this result; the main idea of the proof is to relate the LR-coefficients with canonical bases and total positivity. Lecture III introduces cluster algebras, a new class of commutative algebras defined in [9] in an attempt to create an algebraic framework for canonical bases and total positivity

[1]  Sergey Fomin,et al.  A Littlewood-Richardson Miscellany , 1993, Eur. J. Comb..

[2]  Alek Vainshtein,et al.  Simply laced coxeter groups and groups generated by symplectic transvections , 1999, math/9906203.

[3]  Bernd Sturmfels,et al.  Algorithms in invariant theory , 1993, Texts and monographs in symbolic computation.

[4]  I. Mirkovic,et al.  Perverse Sheaves on affine Grassmannians and Langlands Duality , 1999, math/9911050.

[5]  G. Lusztig.  Braid Group Action and Canonical Bases , 1996 .

[6]  A. Zelevinsky,et al.  Total positivity in Schubert varieties , 1997 .

[7]  Andrei Zelevinsky,et al.  Tensor product multiplicities, canonical bases and totally positive varieties , 1999, math/9912012.

[8]  K. Parthasarathy,et al.  Representations of complex semisimple Lie groups and Lie algebras , 1966 .

[9]  G. Rota,et al.  The invariant theory of binary forms , 1984 .

[10]  S. Fomin,et al.  Parametrizations of Canonical Bases and Totally Positive Matrices , 1996 .

[11]  Sergey Fomin,et al.  The Laurent Phenomenon , 2002, Adv. Appl. Math..

[12]  Totally nonnegative and oscillatory elements in semisimple groups , 1998, math/9811100.

[13]  Sergey Fomin,et al.  Double Bruhat cells and total positivity , 1998, math/9802056.

[14]  S. Fomin,et al.  Cluster algebras I: Foundations , 2001, math/0104151.

[15]  P. Podles,et al.  Introduction to Quantum Groups , 1998 .

[16]  V. Man'ko,et al.  Group Theoretical Methods in Physics , 1987 .

[17]  K. Parthasarathy,et al.  Representations of Complex Semi-Simple Lie Groups and Lie Algebras , 1967 .

[18]  Connected components of real double Bruhat cells , 2000, math/0003231.

[19]  Andrei Zelevinsky,et al.  Canonical bases for the quantum group of type $A_r$ and piecewise-linear combinatorics , 1996 .

[20]  Peter Littelmann,et al.  A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras , 1994 .

[21]  Michael Atiyah,et al.  Lie theory and geometry : in honor of Bertram Kostant , 1997, The Mathematical Gazette.