In the context of algebraic groups, a \(\textbf{maximal torus}\) is a maximal connected abelian subgroup that is diagonalizable over an algebraic closure of the base field. This concept generalizes the notion of maximal tori from Lie groups to algebraic groups, providing a framework for understanding the group's structure and representations.
\(\textbf{Definition:}\)Let \( G \) be a linear algebraic group over a field \( k \). A \(\textbf{torus}\) \( T \) in \( G \) is a subgroup isomorphic to \( (\mathbb{G}_m)^n \) over an algebraic closure \( \overline{k} \), where \( \mathbb{G}_m \) denotes the multiplicative group. A \(\textbf{maximal torus}\) is a torus that is not properly contained in any larger torus within \( G \). (encyclopediaofmath.org)
\(\textbf{Definition. }\)Let \(G\) be a compact simple Lie group with maximal abelian subgroup \(T\) and normalizer \(N(T)\). Then \(W=N(T) / T\) is a finite group called the Weyl group of \(G\).
\(\textbf{Reference. }\)GAY, DAVID A. “CHARACTERS OF THE WEYL GROUP OF SU(n) ON ZERO WEIGHT SPACES AND CENTRALIZERS OF PERMUTATION REPRESENTATIONS.”
\(\textbf{Definition. }\)A finite set \(\Phi \subset \mathbb{E}^d \backslash\{0\}\) is called a crystallographic root system if it spans \(\mathbb{E}^d\) and for all \(\alpha, \beta \in \Phi\), the following hold:
The group \(W\) generated by the reflections \(\sigma_\alpha, \alpha \in \Phi\), is called the Weyl group of \(\Phi\). It is (with a natural choice of generators) a Coxeter group. It is known that every finite irreducible Coxeter group, with the exception of \(H_3, H_4\), and \(I_2(m)\) for \(m \neq 2,3,4,6\), can appear as the Weyl group of a crystallographic root system. The classification of semisimple Lie algebras proceeds via the classification of their root systems and is thus closely linked to the classification of finite Coxeter groups.
\(\textbf{Reference. }\)Combinatorics of Coxeter Groups Björner, Brenti. Page 9.
\(\textbf{Definition. }\)Let \(G\) be a group and \(V, W\) finite dimensional \(G\)-representations. We say that a linear map \(f\) is \(G\)-equivariant if \(f(g * v)=g * f(v)\) forall \(v \in V, g \in G\).
\(\textbf{Remark. }\)The set of all \(G\)-equivariant maps from \(V\) to \(W\) is denoted by \(\text{Hom}_G(V,W)\).
\(\textbf{Definition (A linear functional).}\) A linear functional is a linear map from a vector space to its field of scalars. In mathematical terms, if \( V \) is a vector space over a field \( F \), a linear functional \( f: V \to F \) satisfies: \[ f(ax + by) = af(x) + bf(y), \] for any vectors \( x, y \in V \) and scalars \( a, b \in F \).
A functor is a concept in category theory that maps objects and morphisms from one category to another, preserving the structure of the categories. Specifically, a functor \(\mathcal{F}\) from a category \( \mathcal{C}\) to a category \(\mathcal{D}\) assigns to each object \(X\) in \(\mathcal{C}\) an object \(F(X)\) in \(\mathcal{D}\), and to each morphism \(f: X \to Y\) in \(\mathcal{C}\), a morphism \(\mathcal{F}(f): F(X) \to F(Y)\) in \(\mathcal{D}\). This assignment respects identity morphisms and composition, meaning that \(F(\text{id}_X) = \text{id}_{F(X)}\) and \(F(g \circ f) = F(g) \circ F(f)\).
\(\textbf{Definition (Rational Representation).}\) If \( G \subseteq \mathbb{A}^{N^2} \) is a linear algebraic group, then we call \( (\mathcal{V}, \rho_{\mathcal{V}}) \) a \(\textit{rational representation}\), if \( \rho_{\mathcal{V}} \) is a regular map when regarding \( \mathcal{V} \) as an affine space.
\(\textbf{Definition (Polynomial Representation).}\) A special case are the \(\textit{polynomial representations}\), where we require that \( \rho_{\mathcal{V}} \) is the restriction of a regular function from the affine space \( \mathbb{A}^{N^2} \), which means that all entries in the representation matrices \( \rho_{\mathcal{V}}(g) \) are given by polynomials in the entries of \( g \).
\(\textbf{Reference. }\) Geometric Complexity Theory, Tensor Rank, and Littlewood-Richardson Coefficients.
\(\textbf{Irreducible } GL_n\textbf{-Representations. }\)The irreducible rational representations \(\{ \lambda \}\) of \( GL_n \), sometimes called the Weyl-modules or Schur-modules, are indexed by \( n \)-generalized partitions \( \lambda \). The polynomial representations \(\{ \lambda \}\) are indexed by partitions \( \lambda \vdash n \). Given a \( GL_n \)-representation \( \mathcal{V} \), a vector \( v \in \mathcal{V} \) on which the diagonal matrices act via \[ \text{diag}(\alpha_1, \ldots, \alpha_n)v = \alpha_1^{z_1} \alpha_2^{z_2} \cdots \alpha_n^{z_n} v \] for some \( z \in \mathbb{Z}^n \) is called a \textit{weight vector of weight} \( z \). Every rational \( GL_n \)-representation \( \mathcal{V} \) is a rational representation of the torus \( (\mathbb{C}^\times)^n \) via the embedding as diagonal matrices. Since \( (\mathbb{C}^\times)^n \) is reductive (Thm. 3.3.5), \( \mathcal{V} \) decomposes as a \( (\mathbb{C}^\times)^n \)-representation: \[ \mathcal{V} = \bigoplus_{z \in \mathbb{Z}^n} \mathcal{V}^z, \] where \( \mathcal{V}^z \) denotes the isotypic component of type \( z \). We call \( \mathcal{V}^z \) the \textit{weight space of weight} \( z \), its elements are called \(\textit{weight vectors of weight}\) \( z \), and the decomposition (4.1.1) is called the \(\textit{weight decomposition}\) of \( \mathcal{V} \). For \( GL_3 \), we also have a weight decomposition by choosing the torus \( ((\mathbb{C}^\times)^n)^3 \) and proceeding analogously.
\(\textbf{Reference. }\) Geometric Complexity Theory, Tensor Rank, and Littlewood-Richardson Coefficients.
Suppose \( \phi_1: G \rightarrow \text{GL}\left(V_1\right) \) and \( \phi_2: G \rightarrow \mathrm{GL}\left(V_2\right) \) are representations of a group \( G \). Then we can form a representation \( \phi_1 \otimes \phi_2 \) of \( G \) acting on the tensor product vector space \( V_1 \otimes V_2 \) as follows: \[ \left(\phi_1 \otimes \phi_2\right)(g) = \phi_1(g) \otimes \phi_2(g). \]
\(\textbf{Definition. }\)A representation \( \mathcal{V} \) is called \textit{completely reducible}, if \( \mathcal{V} \) can be written as a direct sum of irreducible \( G \)-subrepresentations.
A linear algebraic group \( G \) is called \textit{linearly reductive}, if every rational \( G \)-representation is completely reducible.
\(\textbf{Reference. }\)Geometric Complexity Theory, Tensor Rank, and Littlewood-Richardson Coefficients.
\(\textbf{Definition (Symmetric Multilinear Function). }\)A multilinear function \( f: U^{\times m} \rightarrow V \) is symmetric if \[ f\left(\vec{v}_1, \ldots, \vec{v}_i, \ldots, \vec{v}_j, \ldots, \vec{v}_m\right) = f\left(\vec{v}_1, \ldots, \vec{v}_j, \ldots, \vec{v}_i, \ldots, \vec{v}_m\right) \] for any \( i \) and \( j \), and for any vectors \( \vec{v}_k \). In other words, for any \(\sigma\in S_m\), we have \[ f\left(\vec{v}_{\sigma(1)}, \ldots, \vec{v}_{\sigma(m)}\right) = f\left(\vec{v}_1, \ldots, \vec{v}_m\right). \]
\(\textbf{Remark. }\)We use \( \text{Sym}\left(U^{\times m}, V\right) \) to denote the set of symmetric multilinear functions from \( U^{\times m} \) to \( V \).
\(\textbf{Definition (Symmetric Power). }\) The \( m^{\text{th}} \) symmetric power of \( V \), denoted \( \text{Sym}^m(V) \), is the quotient of \( V^{\otimes m} \) by the subspace which is generated by \[ \vec{v}_1 \otimes \cdots \otimes \vec{v}_i \otimes \cdots \otimes \vec{v}_j \otimes \cdots \otimes \vec{v}_m - \vec{v}_1 \otimes \cdots \otimes \vec{v}_j \otimes \cdots \otimes \vec{v}_i \otimes \cdots \otimes \vec{v}_m \] where \( i \) and \( j \) and the vectors \( \vec{v}_k \) are arbitrary.
\(\textbf{Theorem. }\)Let \(\{ \vec{v}_1, \ldots, \vec{v}_m \} \) be a basis for \( V \). Then \[ \{ \vec{v}_{i_1} \cdots \vec{v}_{i_n} \mid i_1 \leq \cdots \leq i_n \} \] is a basis for \( \text{Sym}^n(V) \).
\(\textit{Proof. }\)With no restrictions on the subscripts, we can see that \( \{ \vec{v}_{i_1} \otimes \cdots \otimes \vec{v}_{i_n} \} \) form a basis for \( V^{\otimes n} \), the image of these elements spans \( \text{Sym}^n(V) \) (this is equivalent to surjectivity). In the quotient \( \text{Sym}^n(V) \), we can rearrange terms arbitrarily. This lets us rearrange any element of the form \( \vec{v}_{i_1} \cdots \vec{v}_{i_n} \) into one in which the subscripts are in the desired order. For linear independence, we show that there are linear functionals which are non-zero on a chosen \( \vec{v}_{i_1} \cdots \vec{v}_{i_n} \) and zero on all others. This will show that these are linearly independent, by our usual arguments. We can define an equivalence relation on the basis for \( V^{\otimes n} \) by saying that two vectors are equivalent if their subscripts are just permutations of each other. In other words, two basis vectors are equivalent if they map to the same vector in \( \text{Sym}^n(V) \). The equivalence classes form a partition of the basis. It's also clear that a linear functional on \( V^{\otimes n} \) (so a multilinear functional) is symmetric if it takes the same value on equivalent vectors. Since linear functionals on \( V^{\otimes n} \) are determined by their values on a basis, we can easily define a linear functional on \( \text{Sym}^n(V) \) with the required property by considering the linear functional on \( V^{\otimes n} \) which takes value \(1\) on the equivalence class corresponding to our vector and takes value \(0\) on all other equivalence classes. \(\square\)
\(\textbf{Theorem. }\)For \(\text{Sym}^2 V\), we have that \[ \chi_{\text{Sym}^2 V}(g)=\frac{1}{2}\left[\chi_V(g)^2+\chi_V\left(g^2\right)\right] . \] Note that this is compatible with the decomposition \[ V \otimes V=\text{Sym}^2 V \oplus \wedge^2 V . \]
\(\textbf{Reference. }\) Representation Theory : A First Course Page 13.
For \( \lambda = (n) \), the representation \( S^\lambda \) of \( S_n \) is the one-dimensional \textit{trivial representation} \( \mathbb{I}_n \), i.e., the vector space \( \mathbb{C} \) with the action \( \sigma \cdot z = z \) for all \( \sigma \) in \( S_n \) and all \( z \) in \( \mathbb{C} \). The Schur module \( E^{(n)} \) is the \( n^{\text{th}} \) symmetric power \( \operatorname{Sym}^n E \), which is defined to be the quotient space of the tensor product \( E^{\otimes n} \) of \( E \) with itself \( n \) times, dividing by the subspace generated by all differences \[ v_1 \otimes \cdots \otimes v_n - v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(n)} \] for \( v_i \) in \( E \) and \( \sigma \) in \( S_n \). The image of \( v_1 \otimes \cdots \otimes v_n \) in \( \operatorname{Sym}^n E \) is denoted \( v_1 \cdots v_n \). The map \( E^{\times n} \rightarrow \operatorname{Sym}^n E \) is multilinear and symmetric. The vector space \( \operatorname{Sym}^n E \) is determined by a universal property: for any vector space \( F \), and any map \[ \varphi: E^{\times n} \rightarrow F \] that is multilinear and symmetric, there is a unique linear map \[ \tilde{\varphi}: \text{Sym}^n E \rightarrow F \] such that \[ \varphi(v_1 \times \cdots \times v_n) = \tilde{\varphi}(v_1 \cdots v_n). \] \( GL(E) \) acts on \( \text{Sym}^n E \) by \[ g \cdot \left(v_1 \cdot \ldots \cdot v_n\right) = \left(g \cdot v_1\right) \cdot \ldots \cdot \left(g \cdot v_n\right), \] so the symmetric powers are representations of \( GL(E) \). If \( e_1, \ldots, e_m \) is a basis for \( E \), the products \[ e_{i_1} \cdot \ldots \cdot e_{i_n}, \] where the indices vary over weakly increasing sequences \( 1 \leq i_1 \leq i_2 \leq \ldots \leq i_n \leq m \), form a basis for \( \text{Sym}^n E \).
\(\textbf{Reference. }\)Young Tableaux : With Applications to Representation Theory and Geometry. Page 79.
\(\textbf{Remark. }\)When \( \lambda = (n) \), \( E^\lambda \) will be the symmetric product \( \text{Sym}^n E \), and when \( \lambda = \left(1^n\right) \), \( E^\lambda \) will be the exterior product \( \wedge^n E \).
\(\textbf{Reference. }\)Young Tableaux : With Applications to Representation Theory and Geometry. Page 104.
A Young symmetrizer is defined using a Young tableau associated with a partition of \( n \). It involves two subgroups of the symmetric group \( S_n \):
Two elements in the group algebra \( \mathbb{C}S_n \) are constructed:
The Young symmetrizer is the product \( c_{\lambda} = a_{\lambda} b_{\lambda} \), which corresponds to an irreducible representation of \( S_n \).
For instance, if \( n = 4 \) and \( \lambda = (2, 2) \), with the canonical Young tableau \( \{ \{1, 2\}, \{3, 4\} \} \). Then the corresponding \( a_{\lambda} \) is given by \[ a_{\lambda} = e_{\text{id}} + e_{(1,2)} + e_{(3,4)} + e_{(1,2)(3,4)}. \] For any product vector \( v_{1,2,3,4} := v_1 \otimes v_2 \otimes v_3 \otimes v_4 \) of \( V^{\otimes 4} \) we then have \[ v_{1,2,3,4} a_{\lambda} = v_{1,2,3,4} + v_{2,1,3,4} + v_{1,2,4,3} + v_{2,1,4,3} = (v_1 \otimes v_2 + v_2 \otimes v_1) \otimes (v_3 \otimes v_4 + v_4 \otimes v_3). \] Thus the set of all \( a_{\lambda} v_{1,2,3,4} \) clearly spans \( \operatorname{Sym}^2 V \otimes \operatorname{Sym}^2 V \) and since the \( v_{1,2,3,4} \) span \( V^{\otimes 4} \) we obtain \[ V^{\otimes 4} a_{\lambda} \equiv \operatorname{Im}(a_{\lambda}). \] Notice also how this construction can be reduced to the construction for \( n = 2 \). Let \( 1 \in \operatorname{End}(V^{\otimes 2}) \) be the identity operator and \( S \in \operatorname{End}(V^{\otimes 2}) \) the swap operator defined by \( S(v \otimes w) = w \otimes v \), thus \( 1 = e_{\text{id}} \) and \( S = e_{(1,2)} \). We have that \[ e_{\text{id}} + e_{(1,2)} = 1 + S \] maps into \( \operatorname{Sym}^2 V \), more precisely \[ \frac{1}{2} (1 + S) \] is the projector onto \( \operatorname{Sym}^2 V \). Then \[ \frac{1}{4} a_{\lambda} = \frac{1}{4} (e_{\text{id}} + e_{(1,2)} + e_{(3,4)} + e_{(1,2)(3,4)}) = \frac{1}{4} (1 \otimes 1 + S \otimes 1 + 1 \otimes S + S \otimes S) = \frac{1}{2} (1 + S) \otimes \frac{1}{2} (1 + S) \] which is the projector onto \( \operatorname{Sym}^2 V \otimes \operatorname{Sym}^2 V \).
Consider maps \( \varphi: E^{\times \lambda} \rightarrow F \) from \( E^{\times \lambda} \) to an \( R \)-module \( F \), satisfying the following three properties:
1. \( \varphi \) is \( R \)-multilinear. This means that if all the entries but one are fixed, then \( \varphi \) is \( R \)-linear in that entry. 2. \( \varphi \) is alternating in the entries of any column of \( \lambda \). That is, \( \varphi \) vanishes whenever two entries in the same column are equal. Together with (1), this implies that \( \varphi(\mathbf{v}) = -\varphi\left(\mathbf{v}^{\prime}\right) \) if \( \mathbf{v}^{\prime} \) is obtained from \( \mathbf{v} \) by interchanging two entries in a column. 3. For any \( \mathbf{v} \) in \( E^{\times \lambda} \), \( \varphi(\mathbf{v}) = \sum \varphi(\mathbf{w}) \), where the sum is over all \( \mathbf{w} \) obtained from \( \mathbf{v} \) by an exchange between two given columns, with a given subset of boxes in the right chosen column.
\(\textbf{Reference. }\) Young Tableaux : With Applications to Representation Theory and Geometry. Page 105.
\(\textbf{Definition (maximal unipotent group)}\) The group of upper triangular matrices with 1s on the main diagonal is called maximal unipotent group.
\(\textbf{Remark.}\) Usually we use \( U_n \subseteq GL_n \) to denote the maximal unipotent group.
\(\textbf{Definition (Highest Weight Vector)}\) Given a rational \( GL_n \)-representation \( \mathcal{V} \), a weight vector \( v \in \mathcal{V} \) that is fixed under the action of \( U_n \), i.e., \( \forall u \in U_n : u v = v \), is called a \(\textit{highest weight vector}\) of \( \mathcal{V} \).
\(\textbf{Remark.}\) The vector space of highest weight vectors of weight \( \lambda \) is denoted by \( HWV_{\lambda}(\mathcal{V}) \).
\(\textbf{Foulke's Conjecture.}\) Let \(a, b \in \mathbb{N}_{>0}\) with \(a \leq b\). Then for every partition \(\lambda\) the multiplicity of the irreducible \(\text{GL}(V)\) representation \(\{\lambda\}\) in the plethysm \(\text{Sym}^a \text{Sym}^b V\) is at most as large as the multiplicity of \(\{\lambda\}\) in \(\text{Sym}^b \text{Sym}^a V\).
\(\textbf{Remark. }\)We know that \(\text{Sym}^a \text{Sym}^b V\) contains irreducible \(\text{GL}(V)\) representations \(\{\lambda\}\) for which the Young diagram of \(\lambda\) has up to \(a\) rows, but \(\text{Sym}^b \text{Sym}^a V\) contains irreducible \(\text{GL}(V)\) representations \(\{\lambda\}\) for which the Young diagram of \(\lambda\) has up to \(b\) rows.
\(\textbf{Remark. }\)The Foulkes Conjecture is equivalent to saying that there exists a \(\text{GL}(V)\) equivariant inclusion map \[ \text{Sym}^a\text{Sym}^b V \hookrightarrow \text{Sym}^b\text{Sym}^a V. \]
\(\textbf{Reference. }\)Symmetrizing Tableaux and the 5th case of the Foulkes Conjecture