臺灣博碩士論文加值系統

考慮兩個有限維不可分解的sl(n)-module 之張量積。Weyl's Theorem
保證我們可以將該張量積拆解成很多不可分解的子結構之直和。其中，我們將發現，
直和中各項的係數其實就是有名的Littlewood-Richardson coefficients。

Consider the tensor product of two finite dimensional irreducible sl(n)-modules. By
Weyl's Theorem, we can decompose the tensor product into a direct sum of irreducible
sl(n)-submodules. We will prove that the coefficients in the decomposition are actually the
well-known Littlewood-Richardson coefficients.

摘要i
Abstract ii
Contents iii
1 Introduction 1
2 Background knowledge 2
2.1 Denitions and theorems related to representation of Lie algebra . . . . . . 2
2.2 Denitions and theorems related to symmetric group . . . . . . . . . . . . 7
3 Discussions for the sl2 cases. 10
3.1 Decomposing the tensor product of two standard sl2-modules. . . . . . . . 10
3.2 The Clebsch-Gordan formula for sl2 . . . . . . . . . . . . . . . . . . . . . . 12
4 Main results 13
4.1 Clebsch-Gordan formula for sln . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Reformulation in terms of symmetric functions . . . . . . . . . . . . . . . . 13
4.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.4 Littlewood-Richardson Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Bibliography 25

[GW] Roe Goodman Nolan R. Wallach
Symmetry, Representations, and Invariants,
Springer-Verlag, New York, 2009.

[H] Humphreys, J. E.
Introduction to Lie Algebras and Representation Theory,
Springer-Verlag, New York, 1972.

[S] Sagan, B. E.
The Symmetric Group, Representations, Combinatorial Algorithms, and Symmet-
ric Functions,
Springer-Verlag, New York, 2003.