SCHUR FUNCTORS OR THE WEYL CONSTRUCTION MATH 126 FINAL PAPER

ELI BOHMER LEBOW

1. Introduction

We will consider a group G with a representation on a vector space V. The vector space should be a finite dimensional complex vector space. The group need not be finite; later in this paper, the most important example will be G = GL(V). The representation, which will be denoted by just g, v ↦ gv, need not be irreducible for what follows, although that is most important example. Essentially all of the mathematics in this paper is based on [1, pp. 44–47, 75–78, 84–86, 222, 231–232, 471–475]. In addition to these pages, the remaining parts of lectures 4, 6, and 15, and appendices A and B of [1] are also relevant to this topic.

We consider the tensor power,

V^{⊗n} = V ⊗ V ⊗ ⋯ ⊗ V.

We have a representation of G on this space given by g(v_{1} ⊗ ⋯ ⊗ v_{n}) = (gv_{1}) ⊗ ⋯ ⊗ (gv_{n}). Our goal is to break this up into irreducible representations of G. Of course, not knowing anything more about G, we cannot actually do this, but we can break up V^{⊗n} into some subrepresentations, which is progress, at least; and if G = GL(V) these subrepresentations, the Weyl modules, also known as Schur functors, will turn out to be irreducible.

We start by defining some well-known^{1} subspaces of the tensor power. (We will see that these subspaces are invariant under S_{n}.)

Definition 1.1. The nth symmetric power Sym^{n} V of a vector space V is the subspace of V^{⊗n} spanned by

{ ∑_{σ∈S_{n}} v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(n)} | v_{i} ∈ V } .

Definition 1.2. The nth alternating power Alt^{n} V of a vector space V is the subspace of V^{⊗n} spanned by

{ ∑_{σ∈S_{n}} sgn(σ)v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(n)} | v_{i} ∈ V } .

Received by the editors 16 December 1998.

1This means that I had heard of them before taking this course, and is true even though I had originally learned of them from [1, 472–475]

It follows by straightforward calculation that these are invariant subspaces of V^{⊗n}. Except in trivial cases, V^{⊗n} is therefore reducible: if n ≥ 2 and dim V ≥ 2, and v and w are linearly independent elements of V, then v ⊗ ⋯ ⊗ v ∈ Sym^{n} V, so Sym^{n} V is nontrivial, but v ⊗ w ⊗ ⋯ ⊗ w ∉ Sym^{n} V, so Sym^{n} V is a proper subspace of V^{⊗n}.

To find a more obviously generalizable definition of Sym^{n} V and Alt^{n} V, we introduce a representation of S_{n}:

Proposition 1.3. Defining

(v_{1} ⊗ ⋯ ⊗ v_{n})σ = v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(n)} (1)

and extending linearly to the rest of V^{⊗n} yields a (right) representation of S_{n}. This commutes with the (left) representation of G, in the sense that (gω)σ = g(ωσ) for ω ∈ V^{⊗n}.

Proof. It is clear, by direct calculation, that (v_{1}, …, v_{n}) ↦ v_{σ(1)}⊗⋯⊗v_{σ(n)} defines a multilinear map from V × ⋯ × V to V ⊗ ⋯ ⊗ V. By the definition of tensor product (see [1, pp. 471–472]), there is a unique linear map from V ⊗ ⋯ ⊗ V to V ⊗ ⋯ ⊗ V satisfying (1). That (ωσ)τ = ω(στ) for σ, τ ∈ S_{n} is easy calculation when ω = v_{1} ⊗ ⋯ ⊗ v_{n}, and follows by linearity for arbitrary ω ∈ V^{⊗n}. The statement that the two representations commute is just as easy to verify.

With this representation of S_{n}, we can usefully introduce the group algebra. Since the action of G commutes with the action of S_{n}, it also commutes with the action of ℂ[S_{n}]. We can consider V^{⊗n} to be a (right) ℂ[S_{n}]-module, and we are now able to redefine the symmetric and alternating powers:

With this representation of S_{n} in hand, we can usefully introduce the group algebra ℂ[S_{n}]. We can consider V^{⊗n} to be a (right) ℂ[S_{n}]-module. Since the action of G commutes with the action of S_{n}, it also commutes with the action of ℂ[S_{n}]. We are now able to redefine the symmetric and alternating powers:”

Proposition 1.4.

Sym^{n} V = V^{⊗n} · ∑_{σ∈S_{n}} e_{σ}
Alt^{n} V = V^{⊗n} · ∑_{σ∈S_{n}} sgn(σ)e_{σ}

Proof. Since the proofs of the above two equations are so similar, we prove only the second. Let c = ∑_{σ∈S_{n}} sgn(σ)e_{σ}. It follows straight from the definition that Alt^{n} V = Span {(v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(n)})c | v_{i} ∈ V }. We can move the c out of the expression for the set, just by the definition of multiplying a set by an element, and we can move it out the span by linearity of the action of c. Thus, Alt^{n} V = (Span {v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(n)} | v_{i} ∈ V }) c, which is just V^{⊗n}c.

We now see how we might generalize Sym^{n} V and Alt^{n} V. We choose some element c of the group algebra, other than the two used above, and consider V^{⊗n}c.

Proposition 1.5. The subspace V^{⊗n}c of V^{⊗n} is invariant under the action of G, for any c ∈ ℂ[S_{n}].

Proof. Any element of V^{⊗n}c can be expressed as ωc for some ω ∈ V^{⊗n}. Then g(ωc) = (gω)c ∈ V^{⊗n}c, since the two relevant actions commute, so V^{⊗n}c is invariant under G.

We now have an embarrassment of riches. There are uncountably many choices for c. Many choices of different c’s lead to the same V^{⊗n}c, which is good; e.g., if G is finite, V^{⊗n} must break up into only finitely many irreducibles. We can now state our goal more precisely: Find finitely many c such that V^{⊗n} breaks up into a sum of the resulting subrepresentations.

2. The Construction

To clarify the logic of the upcoming construction, we introduce a new kind of tensor product, a tensor product over the algebra ℂ[S_{n}], denoted ⊗_{ℂ[S_{n}]}. We take a right ℂ[S_{n}]-module R, such as V^{⊗n}, and tensor it with a left ℂ[S_{n}]-module L, such as ℂ[S_{n}].

Definition 2.1. Given a right ℂ[S_{n}]-module R, and a left ℂ[S_{n}]-module L, define R ⊗_{ℂ[S_{n}]} L to be R ⊗ L modulo the relation ra ⊗ l ∼ r ⊗ al, for r ∈ R, l ∈ L, and a ∈ ℂ[S_{n}]. (More formally, we quotient out by the relation that generates, i.e., we quotient out by the subspace spanned by {ra ⊗ l − r ⊗ al}.)

Of course, this can be defined over more general algebras than ℂ[S_{n}], and many of the properties are the same, but we only need ℂ[S_{n}].

Proposition 2.2. If R in the above definition is a (left) representation of G, and that representation commutes with the action of ℂ[S_{n}], then R ⊗_{ℂ[S_{n}]} L is also a (left) representation of G, defined by g(r ⊗_{ℂ[S_{n}]} l) = (gr) ⊗_{ℂ[S_{n}]} l.

Proof. The map (r, l) ↦ (gr) ⊗_{ℂ[S_{n}]} l is bilinear, so we have a well-defined linear map r⊗l ↦ (gr)⊗_{ℂ[S_{n}]} l, which sends both ra⊗l and r⊗al to the same element, by the definition of ⊗_{ℂ[S_{n}]} and by the factor the two actions commute. Therefore, the action of g given in the above proposition is well-defined. That it satisfies the other properties of a representation follows straight from the definition of the action in question.

Before we go on, we will prove one useful property of ⊗_{ℂ[S_{n}]}.

Proposition 2.3. The operation ⊗_{ℂ[S_{n}]} distributes over ⊕, in that R ⊗_{ℂ[S_{n}]} (L ⊕ L′) ≅ (R ⊗_{ℂ[S_{n}]} L) ⊕ (R ⊗_{ℂ[S_{n}]} L′), not just as vector spaces but as representations of G.

Proof. To start, we show that R ⊗_{ℂ[S_{n}]} (L ⊕ L′) ≅ (R ⊗_{ℂ[S_{n}]} L) ⊕ (R ⊗_{ℂ[S_{n}]} L′), just as vector spaces. We know that R ⊗ (L ⊕ L′) ≅ (R ⊗ L) ⊕ (R ⊗ L′) as vector spaces. Each of these spaces has to be quotiented out by an equivalence relation; we will write both relations in terms of the latter space, and compare. That is, ω ⊗ a(m ⊕ m′) ∼ ωa ⊗ (m ⊕ m′) maps to the relation (ω ⊗ am) ⊕ (ω ⊗ am′) ∼ (ωa ⊗ m) ⊕ (ωa ⊗ m′) on (R ⊗ L) ⊕ (R ⊗ L′). This space has two relations, one for the first factor and one for the second, which generate the same relation as ω ⊗ am ⊕ ω′ ⊗ a′m′ ∼ ωa ⊗ m ⊕ ω′a′ ⊗ m′. Examining the spaces of vectors identified with zero shows these relations to be the same, so the quotient spaces are isomorphic.

Using this new kind of tensor product, we have another expression for V^{⊗n}c.

Proposition 2.4. V^{⊗n}c ≅ V^{⊗n} ⊗_{ℂ[S_{n}]} ℂ[S_{n}]c, as representations of G.

Proof. First, we need to check that ℂ[S_{n}]c is a left ℂ[S_{n}]-module, before the statement of the proposition even makes sense. But this is easy, since for bc ∈ ℂ[S_{n}]c, we have a(bc) = (ab)c ∈ ℂ[S_{n}]c for any a ∈ ℂ[S_{n}]. That is, ℂ[S_{n}]c is closed under left multiplication by elements of ℂ[S_{n}] so it is a submodule of ℂ[S_{n}] (where we are considering ℂ[S_{n}] as a left-module over itself).

Any element of V^{⊗n} ⊗_{ℂ[S_{n}]} ℂ[S_{n}]c can be expressed as a sum of elements of the form ω ⊗_{ℂ[S_{n}]} bc, which is equal to ωb ⊗_{ℂ[S_{n}]} c. A sum of elements of this form is also of the form ω ⊗_{ℂ[S_{n}]} c. To define our isomorphism, we map this to ωc. Linearity, injectivity, and surjectivity of this map now follow straight from the definitions, provided the map is well-defined. To show this, we consider what the isomorphism does to an element written as ω ⊗_{ℂ[S_{n}]} bc; this is equal to ωb ⊗_{ℂ[S_{n}]} c, and so gets sent to ωbc. We can see by straightforward calculation that map from V^{⊗n} × ℂ[S_{n}]c to V^{⊗n}c defined by (ω, bc) ↦ ωbc is bilinear, so we have a linear map V^{⊗n} ⊗ ℂ[S_{n}]c → V^{⊗n}c. This map sends ωa ⊗ bc and ω ⊗ abc to ωabc, so it is still a well-defined map when we quotient out by the defining relation of ⊗_{ℂ[S_{n}]}.

Since any element of V^{⊗n} ⊗_{ℂ[S_{n}]} ℂ[S_{n}]c can be written in the form ω ⊗_{ℂ[S_{n}]} c, we see that the action of g is just g(ω ⊗_{ℂ[S_{n}]} c) = (gω) ⊗_{ℂ[S_{n}]} c. Our isomorphism clearly respects this.

Having found one submodule of ℂ[S_{n}] useful, we might as well look at them all. Just because S_{n} is a finite group, we know its group algebra breaks up into a sum of (finitely many) irreducible (left) submodules (call them M_{λ}, with λ ranging over some index set to be determined later), with multiplicities equal to their dimensions. Picking some nonzero c_{λ} ∈ M_{λ}, we see that ℂ[S_{n}]c_{λ} is nonzero (since it contains 1c_{λ}), contained in M_{λ} (since that’s a submodule of ℂ[S_{n}]), and in fact equal to M_{λ} (since that’s irreducible). So we now have

ℂ[S_{n}] ≅ ⨁(ℂ[S_{n}]c_{λ})^{⊕m_{λ}} (2)

as ℂ[S_{n}]-modules, with m_{λ} = dim ℂ[S_{n}]c_{λ}. We are now ready to derive the main construction in one big equation. We have

(3) V^{⊗n} ≅ V^{⊗n} ⊗_{ℂ[S_{n}]} ℂ[S_{n}] ≅ V^{⊗n} ⊗_{ℂ[S_{n}]} ⨁(ℂ[S_{n}]c_{λ})^{⊕m_{λ}}

≅ ⨁(V^{⊗n} ⊗_{ℂ[S_{n}]} ℂ[S_{n}]c_{λ})^{⊕m_{λ}} ≅ ⨁(V^{⊗n}c_{λ})^{⊕m_{λ}}.

The first step follows by Proposition 2.4 with c = 1. The other steps can all be justified in an obvious manner from previous propositions. We have thus proven

Theorem 2.5. Using the c_{λ} and m_{λ} that define the irreducibles of S_{n} as in (2), we have V^{⊗n} ≅ ⨁(V^{⊗n}c_{λ})^{⊕m_{λ}} as a sum of subrepresentations.

Note that we haven’t used any particular properties of the representations of S_{n}. A construction of the c_{λ} (where λ ranges over all partitions of n) can be found in [1, 44–47].

3. GL(V)

We now set the group G of the previous sections to be GL(V). This has an natural representation on V, given by gv = g(v). This is irreducible, since given a proper nonzero subspace of V, and given nonzero v in the subspace and w in the complement of the subspace, some element of G satisfies gv = w, so the subspace is not invariant. We can now state the main theorem:

Theorem 3.1. The representations V^{⊗n}c_{λ} of GL(V) are irreducible, unless they are zero.

We break the proof up into lemmas. Due to shortage of time, we omit the proofs.

Lemma 3.2. V^{⊗n} is a left B-module, where B = Sym^{n}(End(V)). This B is exactly the set of linear endomorphisms ϕ of V^{⊗n} which commute with the action of ℂ[S_{n}], in that ϕ(ωc) = ϕ(ω)c for ω ∈ V^{⊗n}, c ∈ ℂ[S_{n}].

Lemma 3.3. An irreducible B-submodule of V^{⊗n} is also an irreducible GL(V)- subrepresentation.

Lemma 3.4. Write V^{⊗n} ≅ ⨁ U_{i}^{⊕ν_{i}}, where the U_{i} are non-isomorphic irreducible representations of S_{n}. Then B is isomorphic (as an algebra) to ⨁ M_{ν_{i}}(ℂ), where M_{ν_{i}}(ℂ) denotes the algebra of ν_{i} × ν_{i} matrices over ℂ.

Lemma 3.5. ℂ[S_{n}] is isomorphic (as an algebra) to ⨁ M_{m_{λ}}(ℂ), with m_{λ} = dim ℂ[S_{n}]c_{λ} being the multiplicity of ℂ[S_{n}]c_{λ} in ℂ[S_{n}].

Lemma 3.6. If ℂ[S_{n}]c_{λ} is an irreducible (left) submodule of ℂ[S_{n}], then V^{⊗n}⊗_{ℂ[S_{n}]} ℂ[S_{n}]c_{λ} is an irreducible (left) B-module.

Using Lemmas 3.3 and 3.6, we have proven the theorem. The result is quite good: we have constructed many irreducible representations of GL(V). In fact (see [1, 4, 231–232] for a fuller statement), we have almost all of the irreducible representations: letting D_{−k} be just ℂ, endowed with the representation in which g acts as (det g)^{−k}, any irreducible representation of GL(V) can be expressed as (V^{⊗n}c_{λ}) ⊗ D_{−k}. (For proof, see [1, Lecture 15, especially p. 232].)

Reference

1. W. Fulton and J. Harris, Representation theory: A first course, Graduate Texts in Mathematics, no. 129, Springer-Verlag, New York, US, 1991.