Lectures in Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 19 August 2013

Lecture 9

Homework Problem 1.60 Let A be a finite dimensional commutative algebra over C. Show that all irreducible representations of A are one dimensional.

Homework Problem 1.61 Let A be a finite dimensional semisimple algebra over C. Show that dimA=λAˆdλ where dλ=dimVλ is the dimension of the irreducible A-module indexed by λAˆ.

2The Symmetric group


I. Basics
II. Characters and Symmetric Functions
III. Construction of Irreducible Modules using Young Symmetrizers.

We won’t do the orthogonal representations of the symmetric group, but we will do them for the Hecke algebras later. The following references will prove useful.

For the orthogonal representations, consult


Definition 2.1 The symmetric group on m letters, 𝒮m, is the group of permutations of the set of integers [1,m] with product defined by composition.

We shall use several equivalent notations for an element π𝒮m.

The symmetric group 𝒮m also has a presentation defined by taking generators {s1,s2,sm-1} subject to the relations (S1) sisj=sjsi |i-j|2 (S2) sisi+1si= si+1sisi+1 1im-2 (S3) si2=1 1im-1 These generators are called simple transpositions. The transposition si corresponds to the permutation (ii+1) written in cycle notation.

We have seen that the trace property implies that characters of a group are constant on conjugacy classes. Furthermore, the irreducible characters are indexed by the same set which indexes the set of conjugacy classes. We thus wish to find the conjugacy classes of the symmetric group 𝒮m.

Let γr=(12r)𝒮m, which is represented by the diagram 1 2 3 4 r-1 r r+1 m For each finite sequence of positive integers μ=(μ1,μ2,,μk), define γμ=γμ1×γμ2××γμk𝒮m where m=i=1kμi. Here γi×γj means the element of 𝒮i+1 obtained by placing the diagram γj to the right of the diagram for γi. For example, if μ=(3,2,1), then γμ𝒮6 is the element γμ= γ3 γ2 γ1

Next we define a procedure that associates to a permutation π𝒮m a sequence of positive integers μ1,μ2,μ3,,μk such that μ1μ2 μ3μk>0 i=1k μi=m Such a sequence of integers is called a partition of m, and we write μm.

To produce a partition from π𝒮m, overlay they identity diagram on the diagram of π. Each edge of the original diagram of π now belongs to exactly one of several disjoint cycles in this new graph. Let μi be the number of edges of the original diagram of π which lie in the ith cycle. For example, given π=(14)(365)𝒮6, we obtain a graph from which we obtain the integers 2, 1, and 3. Arranging these in decreasing order gives the partition (3,2,1) of the number 6.

Define the cycle type of the permutation π to be the partition μ(π) obtained by the above procedure. Observe that given a partition μ, the permutation γμ defined above has cycle type μ.

Proposition 2.2 Let σ𝒮m. Then σ is conjugate to γμ(σ), where μ(σ) is the cycle type of σ.


Write μ=μ(σ)=(μ1,μ2,,μk). Since σ has cycle type μ, the diagram of σ is partitioned into disjoint cycles of lengths μi. It is merely a question of conjugating σ by the permutation that arranges theses cycles in the order in which they appear in γμ.

We obtain this permutation as follows. Recall that the vertices of σ in the top row are labeled from left to right by the integers [1,m]. Let ci,1 denote the label of the leftmost vertex of the top row belonging to the cycle of length μi. To assign values to ci,j for 2jμi, we traverse the cycle of length μi in the diagram of π by proceeding along the edge of π from top to bottom and then jumping from the bottom row to the corresponding vertex in the top row. For 2jμi, let ci,j be the label of the jth vertex visited in the top row of π. Define ρ to be the permutation diagram where the th vertex in the top row is adjacent to the ci,jth vertex in the bottom row where =μ1+μ2++μi-1+j and 1jμi. Observe that this expression uniquely defines i and j and the set of ci,j is in one to one correspondence with [1,m], so ρ is a well-defined permutation diagram. Moreover, ρ-1 is obtained by inverting the diagram of ρ, so that the ci,jth vertex in the top row is adjacent to the =μ1+μ2++μi-1+j vertex of the bottom ρ.

Note that each edge in sigma belongs to exactly one of the cycles of π; hence, a given edge runs from the vertex in the top row labeled by ci,j to the vertex in the bottom row below the vertex labeled by ci,j+1 (where we take ci,μi+1=ci,1). In the composition of the product, therefore, we pass from the vertex in the top row or ρ labeled by =μ1++μi-1+j to the ci,jth vertex in the top row of σ, whence to the ci,j+1 vertex of the top row of ρ-1, which by definition is adjacent to the (+1)st vertex of the bottom row for 1jμi, or to the vertex μ1+μ2++μi-1+1 if j=μi. This results in precisely the diagram of γμ.

As an example, consider the permutation π=(14)(365) with cycle type (3,2,1), above. Then this process results in the product ρπρ-1 = = = γμ

Since the γμ for distinct partitions of m are not conjugate, we have the following corollary.

Proposition 2.3 The conjugacy classes of 𝒮m are indexed by the set of partitions of m.

As usual, we will denote the set of partitions of m by 𝒮mˆ. This is not standard, but agrees with our previously defined notation.

We next introduce some notation and language concerning partitions. Let μ=(μ1,μ2,,μk) be a partition of m. The individual terms of the sequence μi are called the parts of μ. We may use exponential notation to indicate that a certain part is repeated, e.g. (4,2,2,1,1,1) may be written (4,22,13). We identify a partition μ=(μ1,μ2,,μk) with all partitions obtained by padding μ with zeros, i.e. μ=(μ1,μ2,,μk,0n). The length of μ, denoted (μ) is the number of nonzero parts. We will work with partitions of different integers simultaneously, and sometimes will denote the sum of the parts of a partition by |μ|, so μ|μ|.

As always, we work with pictures; for each partition μ, we construct the associated Ferrar’s diagram of μ, which is a frame consisting of m boxes placed as follows. Working top to bottom and left-justified, we place μ1 boxes in the first row, μ2 boxes in the second row, and so on. The boxes are then labeled using matrix notation, i.e. the (i,j) box is the jth box in the ith row from the top. Hence, the partition (6,42,2,14)21 is represented by the Ferrar’s diagram

The conjugate of a partition μ is the partition μ=(μ1,μ2,μr), where μj={μi|μij}. That is, μj is the length of the jth column of μ and hence μ=m. Moreover, the Ferrar’s diagram for μ is the transpose of the Ferrar’s diagram for μ.

Let αi=μi-i and βj=μj-j for 1i(μ) and 1j(μ). Looking at the Ferrars diagram for μ, we see that αi is the number of boxes strictly to the right of the diagonal in the ith row and βj is the number of boxes strictly below the diagonal in the jth column. The Frobenius notation partition, Frobenius notation for for μ is (α1,α2,,α(μ)|β1,β2,,β(μ)).

Let us now identify some representations of the symmetric group. Let V(m):[𝒮m]M1()= be the trivial representation V(m)(σ)=1 for all σ𝒮m. Clearly V(m) is irreducible as it is not zero and has dimension one. The character of this representation χ(m):[𝒮m] is also defined by χ(m)(σ)=1. Hence, the minimal central idempotent corresponding to this representation is given by z(m)=1|G| σ𝒮mdim V(m)χ(m) (σ-1)σ= 1m!σ𝒮mσ

Let us find all one dimensional representations of 𝒮m, which necessarily are irreducible. Suppose ψ:[𝒮m] is a representation. Recall the presentation of the symmetric group 𝒮m defined on generators {s1,s2,,sm-1} defined above. 1=ψ(1)=ψ (si2)=ψ (si)2 so ψ(si)=±1 for each i. With an eye toward relation S2, however, ψ(sisi+1si) = (±1)2ψ (si+1)= ψ(si+1) ψ(si+1sisi+1) = (±1)2ψ (si)=ψ(si). Hence ψ(si)=ψ(sj) for all i and j. If this common value is 1, we have the trivial representation V(m) described above.

Hence, the representation of 𝒮m, V(1m):[𝒮m], defined by V(1m)(si)=-1, is the only other one-dimensional representation besides the trivial representation V(m). The character χ(1m) is called the sign character of 𝒮m.

Finally, we introduce a related algebra.

Definition 2.4 Let q be an indeterminate over and let (q) denote the rational functions in q. The (Iwahori) Hecke algebra, m, is the associative algebra with identity over (q) generated by the elements {g1,g2,,gm-1} subject to the relations (H1) gigj=gjgi for|i-j|2 (H2) gigi+1gi=gi+1gigi+1 1um-2 (H3) gi2=(q-1)gi+q 1im-1

Homework Problem 2.5 Find all one dimensional representations of m.

Notes and References

This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.

page history