# Perspectives in Mathematical Sciences III: Combinatorics and Representation Theory related to Symmetric Groups

## for senior undergraduate students and 1st grd graduate students (English talks, On Monday, 13:00-14:30, Bld 1-109).

## Symmetric Groups

Omidakuji
Outline
## Abstract Algebra (Introductory)

Groups
Rings and Fields
Group Rings
Various homomorphisms (4/18, up to Hom(M,N).)
Category of Modules( conventional )
1-dimensional representations
Restriction and Induction
Permutation Modules
## Combinatorics

Compositions, Partitions, Young Diagrams, Tableaux.
Specht Modules (4/25, up to definition.)
## Refined Inductions and Restrictions

Jucys-Murphy Elements
Central Characters
Grothendieck Groups
## Infinite Dimensional Lie Algebras

sloo-Categorification : QS_n
sl_p^-Categorification : F_p S_n
Hecke algebras
Lascoux-Leclerc-Thibon conjecture

## Reports

1st report: Solve all exercises, write down them on
A4 size papers and submit them.
2nd report: The same task with 1st week plus
the following:
(1) Find definition of a tensor product over a ring.
(2) Find a basis of kS_3 ox_{kS_2} tri.
(3) Using the symmetrizing form,
show the isomorphism kG = Hom_k(kG,k) as bimodules.
(4) Show that our induction is identical with
the tensor induction.
Show the following:
(1) If F_2 is not in k, then we have a ring isomorphism f: kS_2 = k o+ k such that
f is k-linear.
(2) We have a ring isomorphism f : F_2[S_2] = F_2[x]/(x^2) such that f is F_2-linear.
(1) Check the definition of non-split exact sequences.
(2)
Put A=k[x]/(x^2). k is a simple A-module k=k[x]/(x) on which A acts.
Show that
there is a non-split exact sequence
0 -> k -> A -> k -> 0.
sl3. Elements h1,h2,e1,e2,e3,f1,f2,f3 are given in the lecture.
[1] Check that h1,h2,e1,e2,f1,f2 generate sl3 by the Lie bracket.
[2] Check the fundamental relations. [3] Check the definition of Lie algebras
by generators and relations.
Read Section 2 of LLT paper. Write the crystal graph of affine sl_3^, up to |lambda|<7, for the
Kleshchev branching rule.
The differences between LLT paper and ours: (1) LLT used S(lambda) for our Specht module S^{lambda} where the field k contains
the rational field Q.
(2) LLT used S(lambda) with the bar on top for our Specht module S^{lambda} where the field k contains
the finite field F_p.
(3) LLT used the french notation for Young diagrams such as the boxes are piled according to the gravity.
While we used the english notation for Young diagrams such as the boxes are stack to the top roof
contrary to the gravity.