Last update: 19 August 2013
Homework Problem 1.60 Let be a finite dimensional commutative algebra over Show that all irreducible representations of are one dimensional.
Homework Problem 1.61 Let be a finite dimensional semisimple algebra over Show that where is the dimension of the irreducible indexed by
|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 letters, is the group of permutations of the set of integers with product defined by composition.
We shall use several equivalent notations for an element
The symmetric group also has a presentation defined by taking generators subject to the relations These generators are called simple transpositions. The transposition corresponds to the permutation 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
Let which is represented by the diagram For each finite sequence of positive integers define where Here means the element of obtained by placing the diagram to the right of the diagram for For example, if then is the element
Next we define a procedure that associates to a permutation a sequence of positive integers such that Such a sequence of integers is called a partition of and we write
To produce a partition from overlay they identity diagram on the diagram of Each edge of the original diagram of now belongs to exactly one of several disjoint cycles in this new graph. Let be the number of edges of the original diagram of which lie in the cycle. For example, given we obtain a graph from which we obtain the integers 2, 1, and 3. Arranging these in decreasing order gives the partition of the number 6.
Define the cycle type of the permutation to be the partition obtained by the above procedure. Observe that given a partition the permutation defined above has cycle type
Proposition 2.2 Let Then σ is conjugate to where is the cycle type of
Write Since has cycle type the diagram of is partitioned into disjoint cycles of lengths It is merely a question of conjugating by the permutation that arranges theses cycles in the order in which they appear in
We obtain this permutation as follows. Recall that the vertices of in the top row are labeled from left to right by the integers Let denote the label of the leftmost vertex of the top row belonging to the cycle of length To assign values to for we traverse the cycle of length in the diagram of by proceeding along the edge of from top to bottom and then jumping from the bottom row to the corresponding vertex in the top row. For let be the label of the vertex visited in the top row of Define to be the permutation diagram where the vertex in the top row is adjacent to the vertex in the bottom row where and Observe that this expression uniquely defines and and the set of is in one to one correspondence with so is a well-defined permutation diagram. Moreover, is obtained by inverting the diagram of so that the vertex in the top row is adjacent to the vertex of the bottom
Note that each edge in sigma belongs to exactly one of the cycles of hence, a given edge runs from the vertex in the top row labeled by to the vertex in the bottom row below the vertex labeled by (where we take In the composition of the product, therefore, we pass from the vertex in the top row or labeled by to the vertex in the top row of whence to the vertex of the top row of which by definition is adjacent to the vertex of the bottom row for or to the vertex if This results in precisely the diagram of
As an example, consider the permutation with cycle type above. Then this process results in the product
Since the for distinct partitions of are not conjugate, we have the following corollary.
Proposition 2.3 The conjugacy classes of are indexed by the set of partitions of
As usual, we will denote the set of partitions of by This is not standard, but agrees with our previously defined notation.
We next introduce some notation and language concerning partitions. Let be a partition of The individual terms of the sequence are called the parts of We may use exponential notation to indicate that a certain part is repeated, e.g. may be written We identify a partition with all partitions obtained by padding with zeros, i.e. The length of denoted 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 so
As always, we work with pictures; for each partition we construct the associated Ferrar’s diagram of which is a frame consisting of boxes placed as follows. Working top to bottom and left-justified, we place boxes in the first row, boxes in the second row, and so on. The boxes are then labeled using matrix notation, i.e. the box is the box in the row from the top. Hence, the partition is represented by the Ferrar’s diagram
The conjugate of a partition is the partition where That is, is the length of the column of and hence Moreover, the Ferrar’s diagram for is the transpose of the Ferrar’s diagram for
Let and for and Looking at the Ferrars diagram for we see that is the number of boxes strictly to the right of the diagonal in the row and is the number of boxes strictly below the diagonal in the column. The Frobenius notation partition, Frobenius notation for for is
Let us now identify some representations of the symmetric group. Let be the trivial representation for all Clearly is irreducible as it is not zero and has dimension one. The character of this representation is also defined by Hence, the minimal central idempotent corresponding to this representation is given by
Let us find all one dimensional representations of which necessarily are irreducible. Suppose is a representation. Recall the presentation of the symmetric group defined on generators defined above. so for each With an eye toward relation S2, however, Hence for all and If this common value is 1, we have the trivial representation described above.
Hence, the representation of defined by is the only other one-dimensional representation besides the trivial representation The character is called the sign character of
Finally, we introduce a related algebra.
Definition 2.4 Let be an indeterminate over and let denote the rational functions in The (Iwahori) Hecke algebra, is the associative algebra with identity over generated by the elements subject to the relations
Homework Problem 2.5 Find all one dimensional representations of
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.