Last updated: 26 March 2015
This section gives some of the main representation theoretic results used in this thesis, and defines Hecke algebras. Three roughly equivalent structures are commonly used to describe the representation theory of an algebra (1) Modules. Modules are vector spaces on which acts by linear transformations. (2) Representations. Every choice of basis in an vector space on which acts gives a representation, or an algebra homomorphism from to the algebra of matrices. (3) Characters. The trace of a matrix is the sum of its diagonal entries. A character is the composition of the algebra homomorphism of (2) with the trace map. The following discussion will focus on modules and characters.
Let be a finite dimensional algebra over the complex numbers An is a finite dimensional vector space over with a map such that An is irreducible if it contains no nontrivial, proper subspace such that for all and
An homomorphism is a transformation such that and an isomorphism is a bijective homomorphism. Write
Let be an indexing set for the irreducible modules of (up to isomorphism), and fix a set of isomorphism class representatives. An algebra is semisimple if every decomposes In this thesis, all algebras (though not necessarily all Lie algebras) are semisimple.
(Schur's Lemma). Suppose with corresponding irreducible modules and Then
Suppose is a subalgebra of and let be an Then is the vector space over presented by generators with relations Note that the map makes a Define induction and restriction (respectively) as the maps
Suppose is an Every choice of basis of gives rise to an algebra homomorphism The character associated to is The character is independent of the choice of basis, and if are isomorphic then A character is irreducible if is irreducible. The degree of a character is and if then a character is linear; note that linear characters are both irreducible characters and algebra homomorphisms.
Define with a inner product given by Note that the semisimplicity of implies that any character is contained in the subspace If is a character of a subalgebra of then let and if is a character of then is the restriction of the map to
(Frobenius Reciprocity). Let be a subalgebra of Suppose is a character of and is a character of Then
Suppose is a commutative algebra. Then The corresponding character is an algebra homomorphism, and we may identify the label with the character so that
If is a commutative subalgebra of and is an then as a The subspace is the space of and if then has a weight
For large subalgebras such a decomposition can help construct the module For example, 1. If then is a linearly independent set of vectors. In particular, if for all then is a basis for 2. If is irreducible, then Chapter 6 uses this idea to construct irreducible modules of the Yokonuma algebra.
Let be a finite group. The group algebra is the algebra with basis and multiplication determined by the group multiplication in A is a vector space over with a map such that Note that there is a natural bijection so I will use and interchangeably. Let index the irreducible
The inner product (2.1) on has an explicit expression and if is a subgroup of then induction is given by
An idempotent is a nonzero element that satisfies For let These are the minimal central idempotents of and they satisfy In particular, as a
Let be a subgroup of with an irreducible The Hecke algebra is the centralizer algebra If is an idempotent such that then and the map is an algebra anti-isomorphism (i.e.
The following theorem connects the representation theory of to the representation theory of Theorem 2.3 and the Corollary are in [CRe1981, Theorem 11.25].
(Double Centralizer Theorem). Let be a and let Then, as a if and only if, as an where is the set of irreducible
Suppose is a subgroup of Let be an irreducible Write Then the map is a bijection that sends for every irreducible that is isomorphic to a submodule of
There are two common approaches used to define finite Chevalley groups. One strategy considers the subgroup of elements in an algebraic group fixed under a Frobenius map (see [Car1985]). The approach here, however, begins with a pair of a Lie algebra and a and constructs the Chevalley group from a variant of the exponential map. This perspective gives an explicit construction of the elements, which will prove useful in the computations that follow (see also [Ste1967]).
A finite dimensional Lie algebra is a finite dimensional vector space over with a map such that (a) for all (b) for Note that if is any algebra, then the bracket for gives a Lie algebra structure to
A is a vector space with a map such that A Lie algebra acts diagonally on if there exists a basis of such that
The center of is Let index the irreducible A Lie algebra is reductive if for every finite-dimensional module on which acts diagonally, If is reductive, then
Let be a reductive Lie algebra. A Cartan subalgebra of is a subalgebra satisfying (a) for some is nilpotent), (b) is its own normalizer). If is a Cartan subalgebra of then is a Cartan subalgebra of Let
As an decomposes The set of roots of is A set of simple roots is a subset of the roots such that (a) with linearly independent, (b) Every root can be written as with either or Every choice of simple roots splits the set of roots into positive roots and negative roots In fact,
Example. Consider the Lie algebra and bracket given by Write where and Also, Let be the simple root given by so that and
For every pair of roots there exists a Lie algebra isomorphism Under this isomorphism, let Note that The following classical result can be found in [Hum1972, Theorem 25.2].
(Chevalley Basis). There is a choice of the such that the set is a basis of satisfying (a) is a combination of (b) Let such that and suppose are maximal such Then
A basis as in Theorem 2.5 is a Chevalley basis of (For an analysis of the choices involved see [Sam1969] and [Tit1966]).
Let be a finite dimensional such that has a that satisfies (a) There exists a of such that (1) (2) for all and (3) (b) for and (c) (Condition (a) guarantees that acts diagonally. If then the existence of such a basis is guaranteed by a theorem of Kostant [Hum1972, Theorem 27.1]).
Let be a finite dimensional that satisfies (a)-(c) above. For let As a transformation of the basis the element with Let Temporarily abandon precision (i.e. allow infinite sums) to obtain This computation motivates some of the definitions below.
Let The finite field with elements has a multiplicative group and an additive group Let
The finite reductive Chevalley group is where
Remarks. 1. If we “allow” infinite sums, then 2. If then
Example. Suppose (as before) and let be the natural given by matrix multiplication. Then has a basis By direct computation, and (the general linear group).
This section describes some of the pertinent combinatorial objects and techniques.
Let be the simple reflection that switches and and fixes everything else. The group can be presented by generators and relations Using two rows of vertices and strands between the top and bottom vertices, we may pictorially describe permutations in the following way. View Multiplication in corresponds to concatenation of diagrams, so for example, Therefore, is generated by with relations
A composition is a sequence of positive integers. The size of is the length of is and If then is a composition of and we write View as a collection of boxes aligned to the left. If a box is in the row and column of then the content of is For example, if then and the contents of the boxes are Alternatively, coincides with the numbers in the boxes at the end of the rows in the diagram
A partition is a composition where If then is a partition of and we write Let Suppose The conjugate partition is given by In terms of diagrams, is the collection of boxes obtained by flipping across its main diagonal. For example,
Suppose are partitions. If for all then the skew partition is the collection of boxes obtained by removing the boxes in from the upper left-hand corner of the diagram For example, if A horizontal strip is a skew shape such that no column contains more than one box. Note that if then is a partition.
A column strict tableau of shape is a filling of the boxes of by positive integers such that (a) the entries strictly increase along columns, (b) the entries weakly increase along rows. The weight of is the composition given by For example,
The symmetric group acts on the set of variables by permuting the indices. The ring of symmetric polynomials in the variables is For the elementary symmetric polynomial is and the power sum symmetric polynomial is For any partition let The Schur polynomial corresponding to is for which Pieri’s rule gives
For each the Hall-Littlewood symmetric function is where is a constant as in [Mac1995, III.2]; and they satisfy For and if we extend coefficients to then
Note that for there is a map which sends etc. Let be an infinite set of variables. The Schur function is the infinite sequence and one can define elementary symmetric functions power sum symmetric functions and Hall-Littlewood symmetric functions analogously. The ring of symmetric functions in the variables is and let
The classical RSK correspondence provides a combinatorial proof of the identity by constructing for each a bijection between the matrices and the set of pairs of column strict tableaux with the same shape. The bijection is as follows.
If is a column strict tableau and let be the column strict tableau given by the following algorithm (a) Insert into the the first column of by displacing the smallest number If all numbers are then place at the bottom of the first column. (b) Iterate this insertion by inserting the displaced entry into the next column. (c) Stop when the insertion does not displace an entry.
A two-line array is a two-rowed array with and if If then let be the two-line array with pairs
For suppose Then the pair is the final pair in the sequence where is a pair of column strict tableaux with the same shape given by For example, and provides the sequence so that
This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.