DUALITY FOR THE SPIN REPRESENTATION

Jeffrey Adams

Consider a reductive dual pair (G_{m}, G_{n}) in SO(mn). Lifting to Spin(mn), and restricting the spin representation we get a bijection between finite dimensional representations of appropriate covers of G_{m} and G_{n}. Here is the explicit correspondence.

§1. Orthogonal Groups

Recall the definition of the groups Spin(m) → SO(m) and P in(m) → O(m) ([.cartan.], [.broecker.]). We parametrize irreducible representations of these groups by triples D = (Y, δ, ϵ) where Y is a Young’s diagram with number of rows less than or equal to the rank of the orthogonal group, δ ∈ \frac{1}{2}Z/Z and ϵ ∈ ±1. We first consider representations of Spin(m). Let t = [\frac{m}{2}], and for δ ∈ \frac{1}{2}Z let \overline{δ} ∈ \{0, \frac{1}{2}\} be the corresponding coset representative.

The triple (Y, δ, 1), where Y has rows r_{1}, …, r_{t}, corresponds to the irreducible representation of Spin(m) with highest weight (r_{1} + \overline{δ}, …, r_{t} + \overline{δ}). This factors to SO(m) if and only if δ = 0. If m is odd ϵ has no effect. If m is even, ϵ has no effect unless r_{t} ≠ 0, in which case (Y, δ, −1) gives the represesentation with highest weight (r_{1} + \overline{δ}, …, r_{t−1} + \overline{δ}, −r_{t} − \overline{δ}).

Now consider P in(m), which is a subgroup of the Clifford algebra of R^{m}, and contains Spin(m) as a subgroup of index two. If m is odd, P in(m) is generated by Spin(m) and α = e_{1}e_{2} … e_{m}; α is central, and α^{2} = 1 (resp. −1) if m ≡ 3(4) (resp. 1(4)). Therefore in this case P in(m) is a central extension of Spin(m), which splits iff m ≡ 3(4). If m is even this is a non–split, non–central extension.

The parametrization of representations of P in(m) in terms of those of Spin(m) is analogous to the situation with O(m) and SO(m). If m is even D = (Y, δ, ϵ) corresponds to the representation of P in(m) with highest weight (r_{1} +\overline{δ}, …, r_{t} +\overline{δ}). If r_{t} > 0, ϵ plays no role and this representation restricts to a direct sum of two irreducible representations of Spin(m) with highest weights (r_{1} + \overline{δ}, …, r_{n−1} + \overline{δ}, ±(r_{t} + \overline{δ})). If r_{t} = 0 there are two representations of Spin(m) given by ϵ = ±1, each of which is irreducible when restricted to Spin(m). If m is odd there are always two irreducible representations of P in(m) given by ϵ = ±1, having the same restriction to Spin(m). Of course all of these results are obtained by inducing from Spin(m) to P in(m).

Lemma 1.1.

The irreducible representations of Spin(m) are parametrized by data D = (Y, δ, ϵ). The representations given by (Y, δ, ϵ) and (Y, δ, −ϵ) are isomorphic unless m is even and r\frac{m}{2} + \overline{δ} > 0.

Typeset by AMS-TEX

The irreducible representations of P in(m) are parametrized by D = (Y, δ, ϵ). In this setting ϵ and −ϵ give distinct representations unless m is even and r\frac{m}{2} + \overline{δ} > 0.

For any of these group let π(D) be the corresponding representation.

Definition 1.2. Fix m and n and suppose Y is a Young’s diagram with less than or equal to m rows and n columns. Embed Y in a rectangle R of size m × n. The complement Y^{c} of Y is the diagram obtained by taking the boxes of R not contained in Y. The transpose ^{t} is the usual transpose about the diagonal. Then the Young’s diagram Ỹ is defined by taking (Y^{c})^{t} = (Y^{t})^{c} and rearranging the rows in decreasing order.

Thus if Y has columns c_{1}, c_{2}, …, c_{n}, Ỹ has rows m − c_{n}, m − c_{n−1}, …, m − c_{1}. For example if m = 4, n = 5 and Y = (4, 3, 1) (i.e. rows 4, 3 and 1) then Ỹ = (4, 3, 2, 2, 1).

Now consider V = R^{m} and V′ = R^{n} with positive definite symmetric bilinear forms, and W = V ⊗ V′ with the product form. We denote the symmetry groups O(m), O(n) and O(mn). There is a natural map O(m) × O(n) → O(mn). The corresponding statement for the spin group is a bit subtle.

Lemma 1.3. There are natural maps ϕ:

(1) ϕ : Spin(m) × P in(n) → Spin(mn) m, n even;

(2) ϕ : Spin(m) × Spin(n) → Spin(mn) m, n odd;

(3) ϕ : Spin(m) × P in(n) → Spin(mn) m even, n odd.

(There are also natural maps P in(m) × P in(n) → P in(mn) in case both m and n are odd, and P in(m) × Spin(n) → P in(mn) if m is even and n is odd.)

proof. Let u_{1}, … u_{m} be an orthonormal basis of V, and similarly v_{1}, … v_{n} for V′. Then {z_{i,j} = u_{i} ⊗ v_{j}} is an orthonormal basis of W. The map ϕ : u_{i} → ∏_{k=1}^{n} z_{ik} defines a map P in(m) → P in(mn). Similarly for P in(n). Then an easy calculation shows:

ϕ(u_{i})ϕ(v_{j}) = (−1)^{mn−1}ϕ(v_{j})ϕ(u_{i}).

Note that the images of P in(m) and P in(m) commute iff mn is odd. Also note that the images of Spin(m) and P in(n) commute, and vice versa. It is also immediate that the image of Spin(m) is contained in Spin(mn), whereas ϕ takes P in(m) to Spin(mn) only if n is even. The result follows.

In the setting of Lemma 1.3 we let (G_{m}, G_{n}) be the groups appearing in the Lemma, with map ϕ : G_{m} × G_{n} → Spin(mn). Let ω be the spin representation of Spin(mn). This has dimension 2^{[ \frac{mn}{2} ]}, is irreducible if mn is odd, and is the direct sum of two irreducible summands of dimension 2^{[ \frac{mn}{2} ]−1} if mn is even. We consider ω as a representation of G_{m} × G_{n} via the map ϕ. Then ω ≃ ∑_{i} π_{i} ⊗ π′_{i} as a representation of (G_{m}, G_{n}), and ([.howe remarks.]) the correspondence π_{i} → π′_{i} is a bijection.

Theorem 1.4.

Let D = (Y, \frac{n}{2}, ϵ) be data such that Y has less than or equal to [\frac{m}{2}] rows and [\frac{n}{2}] columns. Let D′ = (Ỹ , \frac{m}{2}, ϵ). Then the representations π(D) of G_{m} and π(D′) of G_{n} correspond to each other, and all corresponding representations are obtained this way.

Comments and Examples.

For Spin(n) and P in(n) we write representations in terms of highest weights, with ϵ as a subscript: (a_{1}, a_{2}, …, a_{[ \frac{n}{2} ]})_{ϵ}. We omit ϵ if it is unnecessary. Also for Spin(n) we write (a_{1}, a_{2}, …, a_{[ \frac{n}{2} ]})_{±} = (a_{1}, a_{2}, …, ±a_{[ \frac{n}{2} ]}).

For the pair (O(m), O(n)) all highest weights λ = (a_{1}, a_{2}, …, a_{[ \frac{m}{2} ]})_{ϵ} for Spin(m) or P in(m) occur with \frac{n}{2} − a_{i} ∈ ℕ, and all ϵ.

Theorem 1.4 may also be stated with the alternative embeddings into P in(mn) following Lemma 1.3.

The representations of Spin(m) and P in(m) are genuine if and only if n is odd, and vice versa. Note that if n is odd, the weights for Spin(m) are half-integers, and in particular non–zero, so both signs ϵ always occur. This corresponds to all signs ϵ occuring for Spin(n) for n odd.

(1):(P in(1), Spin(2n)):

P in(1) is isomorphic to Z/4Z, and has two non–genuine representations (0)_{±}. These go to the two halves of the spin representation of Spin(2n):

(0)_{±} → (\frac{1}{2}, …, ±\frac{1}{2}).

This may also be stated in terms of (Spin(1), P in(2n)): The spin module of P in(2n) is irreducible, corresponding to the unique non–genuine character of the group Spin(1) ≃ Z/2Z.

(2): (Spin(1), Spin(2n + 1)): The spin module for Spin(2n + 1) is irreducible, corresponding to the fact that Spin(1) = Z/2Z has a unique genuine representation:

(0) → (\frac{1}{2}, …, \frac{1}{2}).

(3): One dimensional representations. If n is even the one–dimensional representation(s) of Spin(m) or P in(m) occur in the correspondence. The corresponding representation(s) are (\frac{m}{2}, …, \frac{m}{2})_{ϵ}. More precisely:

(1) Spin(2m), P in(2n): (0) → (m, m, …, m),

(2) P in(2m), Spin(2n): (0)_{±} → (n, n, …, ±n),

(3) P in(2m + 1), Spin(2n): (0)_{±} → (m + \frac{1}{2}, m + \frac{1}{2}, …, ±(m + \frac{1}{2})).

(4): P in(2), Spin(2n):

The non–genuine representations of P in(2) are {(k)|k = 1, 2, … }∪{(0)_{±}}, where (k) is two–dimensional and (0)_{±} are one–dimensional. Then:

(1) (0)_{±} → (1, 1, …, 1, ±1),

(2) (k) → (1, 1, …, 1, \overbrace{0, 0, …, 0}^{k}) k = 1, 2, …, n.

(5): P in(3), Spin(2n):

In this case all the occuring representations of Spin(2n) are genuine, and both signs ϵ = ±1 occur. The representations of P in(3) are non–genuine, and also both signs occur. The correspondence is:

(k)_{±} → (\frac{3}{2}, \frac{3}{2}, …, \frac{3}{2} \overbrace{\frac{1}{2}, \frac{1}{2}, …, \frac{1}{2}}^{k})_{±} k = 0, 1, …, n.

Proof of Theorem 1.4.

We restrict ourselves to the case O(2m), O(2n). Choose coordinates {e_{i,j}, f_{i,j}|1 ≤ i ≤ 2m, 1 ≤ j ≤ n} on R^{4mn} such that (e_{i,j}, e_{k,ℓ}) = (f_{i,j}, f_{k,ℓ}) = 0 and (e_{i,j}, f_{k,ℓ}) = δ_{i,k}δ_{j,ℓ}. Then the Clifford algebra C of R^{4mn} is generated by the e_{i,j} and f_{i,j} with the usual relations [.lawson.]. The spin representation of Spin(4mn) is then realized on the subalgebra of C generated by {e_{i,j}}, and 𝔰𝔬(4mn) is the quadratic elements in C.

The following operators, with 1 ≤ i, j ≤ m form a subalgebra of 𝔰𝔬(4mn) ⊂ C isomorphic to 𝔰𝔬(2m):

(1) X_{i,j}^{+} = ∑_{k=1}^{n}(e_{i,k}e_{j+m,k} − e_{j,k}e_{i+m,k});

(2) X_{i,j}^{−} = ∑_{k=1}^{n}(f_{i,k}f_{j+m,k} − f_{j,k}f_{i+m,k});

(3) Z_{i,j} = ∑_{k=1}^{n}(e_{i,k}f_{j,k} + e_{i+m,k}f_{j+m,k} − δ_{i,j}). A Cartan subalgebra is spanned by the Z_{i,i}, and the nilpotent part of a Borel subalgebra by {Z_{i,j}|i < j} ∪ {X_{i,j}^{+}}.

It is not hard to compute the commutant of this algebra in 𝔰𝔬(4mn). It is contained in the sublagebra generated by the elements e_{i,j}f_{k,ℓ}, which is isomorphic to 𝔤𝔩(2mn). Let 1, ≤ i, j ≤ n:

(1) W_{i,j} = ∑_{k=1}^{m}(e_{k,i}f_{k,j} − e_{k+m,j}f_{k+m,i});

(2) W_{i,j}^{+} = ∑_{k=1}^{m}(e_{k,i}f_{k+m,j} − e_{k,j}f_{k+m,i});

(3) W_{i,j}^{−} = ∑_{k=1}^{m}(f_{k,i}e_{k+m,j} − f_{k,j}e_{k+m,i}).

A Cartan subalgebra is spanned by the W_{i,i}, and the nilpotent part of a Borel subalgebra by {W_{i,j}|i < j} ∪ {W_{i,j}^{+}}.

For any non–negative integers m ≥ α_{n} ≥ ⋯ ≥ α_{1} consider the following diagram: (1.5)

e_{1,n} … e_{m,n}  e_{m+1,n}  …  …  …  e_{m+α_{n},n}
e_{1,n−1} … e_{m,n−1} e_{m+1,n−1}  …  …  e_{m+α_{n−1},n−1}
⋮  ⋮  ⋮  ⋮  ⋮  ⋮
e_{1,3} … e_{m,3}  e_{m+1,3}  …  e_{m+α_{2},3}
e_{1,2} … e_{m,2}  e_{m+1,2}  …  e_{m+α_{2},2}
e_{1,1} … e_{m,1}  e_{m+1,1} e_{m+α_{1},1}

Let ψ be the product of all the terms occuring in the diagram. Then ψ is readily seen to be annihlated by the nilpotent parts of the Borel subalgebras above. Furthermore Z_{i,i} acts by β_{i} where β_{i} is the height of the m + i-th column, and W_{j,j} acts by m−α_{j}. Therefore the weights (β_{1}, β_{2}, …, β_{m}) and (n−α_{1}, n−α_{2}, …, n−α_{n}) correspond. The shape of the Young’s diagram Y is the shape of the right half of (1.5). This gives the corresponding representations for ϵ ≠ 0.

Furthermore β_{m} ≠ 0 if and only if n − α_{n} = 0. In this case there is an additional representation coming from ϵ = −1. This highest weight vector is found by deleting the m-th column of the diagram.

As a representation of Spin(2m), ω is the n–fold tensor product of the standard module with itself. From this is it clear that we have obtained all possible highest weights of Spin(2m) modules occuring in ω. This proves this is the complete correspondence.

§2. General Linear Groups

Consider the dual pair (GL(m), GL(n)) ⊂ O(2mn). In this case the pairing is very simple. The covering group G̃L(m) of GL(m) is the det^{\frac{n}{2}} cover. This cover has a distinguished character det^{\frac{n}{2}} , and tensoring with this character gives a bijection between the irreducible representations of GL(m) and G̃L(m). Let Y be a Young’s diagram with less than or equal to m rows, with associated representation π(Y) of GL(m). Let π̃(Y) be the representation π(Y) ⊗ det^{− \frac{n}{2}} of G̃L(m). Similarly for G̃L(n), interchanging the roles of m and n.

Theorem 2.1. Let Y be a Young’s diagrams with less than or equal to m rows and n columns. Then π̃(Y) corresponds to π̃(Y^{t}), and all corresponding representations are obtained this way.

proof. We use coordinates {e_{i,j}, f_{i,j}|1 ≤ i ≤ m, 1 ≤ j ≤ n}. Then 𝔤𝔩(m) acts by ∑_{k=1}^{n} e_{i,k}f_{j,k} − \frac{n}{2}δ_{i,j} (1 ≤ i, j ≤ m). Similarly 𝔤𝔩(n) acts by ∑_{k=1}^{m} e_{k,i}f_{k,j} − \frac{m}{2}δ_{i,j} (1 ≤ i, j ≤ n). Then for n ≥ α_{1} ≥ α_{2} ⋯ ≥ α_{m}, taking the product of the factors in the diagram:

e_{1,1} e_{2,1} . . . . . .  … e_{α_{1},1}
e_{1,2} e_{2,2} . . . . . . e_{α_{2},2}
⋮  ⋮  ⋮  ⋮
e_{1,m} e_{2,m} … e_{α_{m},m}

gives a highest weight vector for both groups, with the indicated highest weights.