DERANGEMENTS AND TENSOR POWERS

OF ADJOINT MODULES FOR 𝔰𝔩_{n}

Georgia Benkart^{1} Stephen Doty June 30, 2001

Abstract. We obtain the decomposition of the tensor space 𝔰𝔩_{n}^{⊗k} as a module for 𝔰𝔩_{n}, find an explicit formula for the multiplicities of its irreducible summands, and (when n ≥ 2k) describe the centralizer algebra C = End_{𝔰𝔩_{n}} (𝔰𝔩_{n}^{⊗k}) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of C is given by the number of derangements of a set of 2k elements.

Introduction

Weyl’s celebrated theorem on complete reducibility says that a finite-dimen-sional module X for a finite-dimensional simple complex Lie algebra 𝔤 is a direct sum of irreducible 𝔤-modules. However, to determine an explicit expression for the multiplicities of the irreducible 𝔤-summands of X often is a very challenging task. In this note we assume 𝔤 = 𝔰𝔩_{n}, the simple Lie algebra of n × n matrices of trace 0 over ℂ, and view 𝔰𝔩_{n} as a 𝔤-module under the adjoint action x · y = [x, y]. We take X to be the k-fold tensor power of 𝔰𝔩_{n}. Using combinatorial methods and results developed in [BCHLLS], we establish an explicit description of the irreducible 𝔤-summands of 𝔰𝔩_{n}^{⊗k} (Theorem 1.16) and determine an expression for their multiplicities (Theorem 2.2). As a consequence of our formula, we obtain the following results, expressed in terms of the number D_{k} of derangements of {1, …, k}: For n ≥ 2k, the dimension of the space of 𝔤-invariants in 𝔰𝔩_{n}^{⊗k} is D_{k}; the multiplicity of 𝔰𝔩_{n} in 𝔰𝔩_{n}^{⊗k} is D_{k+1}; and the dimension of the centralizer algebra C = End_{𝔤}(𝔰𝔩_{n}^{⊗k}) is D_{2k}.

In Section 3, we identify the centralizer algebra C with a certain subalgebra of the walled Brauer algebra B_{k,k}(n). This subalgebra has a basis indexed by

Typeset by AMS-TEX

1Supported in part by NSF Grant #9970119 2000 Mathematical Subject Classification Primary 17B10, Secondary 05E10

derangements of {1, …, 2k}. We then give a description (for n ≥ 2k) of the irreducible modules for C, and obtain the “double centralizer” decomposition of the tensor space 𝔰𝔩_{n}^{⊗k} as a bimodule for C × 𝔤.

§1. The Tensor Product Realization

The general linear Lie algebra 𝔤𝔩_{n} = 𝔰𝔩_{n} ⊕ ℂI of all n × n complex matrices acts on 𝔰𝔩_{n} via the adjoint action, and the identity matrix I acts trivially. Hence, there is no harm in assuming that 𝔤 is 𝔤𝔩_{n} rather than 𝔰𝔩_{n} acting on 𝔰𝔩_{n}^{⊗k} in what follows; the results are exactly the same. This enables us to label the irreducible summands by pairs of partitions and to apply known results on the decomposition of tensor products for 𝔤𝔩_{n}.

Let 𝔥 denote the Cartan subalgebra of 𝔤 = 𝔤𝔩_{n} of diagonal matrices, and let ϵ_{i} : 𝔥 → ℂ be the projection of a diagonal matrix onto its (i, i)-entry. The irreducible finite-dimensional 𝔤-modules are labeled by their highest weight, which is an integral linear combination ∑_{i=1}^{n} κ_{i}ϵ_{i} with κ_{1} ≥ κ_{2} ≥ ⋯ ≥ κ_{n}. By letting λ = (λ_{1} ≥ λ_{2} ≥ ⋯) denote the sequence of positive κ_{i} and μ = (μ_{1} ≥ μ_{2} ≥ ⋯) be the partition determined by the negative κ_{i}, we may associate to each highest weight a pair of partitions (λ, μ). For example, for 𝔤 = 𝔤𝔩_{12} the highest weight

3ϵ_{1} + 2ϵ_{2} + 2ϵ_{3} + 2ϵ_{4} + ϵ_{5} − 4ϵ_{10} − 5ϵ_{11} − 5ϵ_{12}

is identified with the pair of partitions λ = (3, 2, 2, 2, 1) ⊢ 10 and μ = (5, 5, 4) ⊢ 14. Therefore, the set of highest weights for 𝔤-modules is in bijection with the set of pairs of partitions such that the total number of nonzero parts does not exceed n.

Let V = ℂ^{n} be the natural representation of 𝔤 = 𝔤𝔩_{n} on n × 1 matrices by matrix multiplication. The dual module V^{∗} may be identified with 1 × n matrices, where the 𝔤-action is by right multiplication by the negative of an element x ∈ 𝔤. The matrix product

V ⊗ V^{∗} → 𝔤𝔩_{n} = 𝔰𝔩_{n} ⊕ ℂI,  u ⊗ w^{∗} ↦ uw^{∗}  (1.1)

is a 𝔤-module isomorphism which allows us to identify 𝔤𝔩_{n} with V ⊗ V^{∗}.

Let {v_{1}, …, v_{n}} denote the standard basis of V, where v_{i} is the matrix having 1 in the ith row and 0 everywhere else. Assume {v_{1}^{∗}, …, v_{n}^{∗}} is the dual basis in V^{∗}, so that v_{i}^{∗} has 1 in its ith column and 0 elsewhere. The contraction mapping c : V ⊗ V^{∗} → V ⊗ V^{∗} is defined using the trace by

c(u ⊗ w^{∗}) = tr(uw^{∗}) ∑_{ℓ=1}^{n} v_{ℓ} ⊗ v_{ℓ}^{∗}.  (1.2)

DERANGEMENTS AND TENSOR POWERS OF 𝔰𝔩_{n} 3

Under the isomorphism in (1.1), v_{ℓ}⊗v_{ℓ}^{∗} is mapped to the matrix unit E_{ℓ,ℓ} ∈ 𝔤𝔩_{n}. Therefore, we may identify the image of c with ℂI, and the kernel of c with 𝔰𝔩_{n}.

As c^{2} = nc, the mapping p = (1/n)c is an idempotent. It is the projection onto the trivial summand ℂI, and id − p is the projection onto 𝔰𝔩_{n}. These idempotents are orthogonal,

p(id − p) = 0 = (id − p)p,

and satisfy id = (id − p) + p. (Here id is the identity map on V ⊗ V^{∗}.)

In order to identify 𝔰𝔩_{n}^{⊗k} with a summand of

M = V^{⊗k} ⊗ (V^{∗})^{⊗k} ≅ (V ⊗ V^{∗})^{⊗k} ≅ 𝔤𝔩_{n}^{⊗k},  (1.3)

we define the contraction map c_{i,j} to be the contraction c applied to the ith factor of V^{⊗k} and the jth factor of (V^{∗})^{⊗k} according to

c_{i,j}(u_{1} ⊗ ⋯ ⊗ u_{k} ⊗ w_{1}^{∗} ⊗ ⋯ ⊗ w_{k}^{∗})

= tr(u_{i}w_{j}^{∗}) ∑_{ℓ=1}^{n} u_{1} ⊗ ⋯ v_{ℓ} ⊗ ⋯ ⊗ v_{k} ⊗ w_{1}^{∗} ⊗ ⋯ ⊗ v_{ℓ}^{∗} ⋯ ⊗ w_{k}^{∗},

where v_{ℓ} is placed in the ith slot of V^{⊗k} and v_{ℓ}^{∗} in the jth slot of (V^{∗})^{⊗k}. As before, c_{i,j}^{2} = nc_{i,j}, so that

p_{i} = \frac{1}{n}c_{i,i}  (1.4)

is an idempotent.

Proposition 1.5. ker p_{1}∩ker p_{2}∩⋯∩ker p_{k} = (id−p_{1})(id−p_{2}) ⋯ (id−p_{k})M.

Proof. The idempotents p_{i} commute and satisfy p_{i}(id−p_{i}) = 0. For J a subset of {1, …, k}, let p_{J} = ∏_{j∈J} p_{j}. Set q_{j} = id − p_{j} and q_{J} = ∏_{j∈J} q_{j}. Then

M = ⨁_{J⊆{1,...,k}} p_{J^{c}}q_{J}M

This can be argued by induction on k. Note that the sum is direct because for any fixed choice of subset J′, the idempotent p_{J′}^{c}q_{J′} acts as the identity on p_{J′}^{c}q_{J′}M and annihilates the remaining terms p_{J^{c}}q_{J}M with J ≠ J′. Whenever j ∈ J^{c}, then p_{J^{c}}q_{J}M is not contained in ker p_{j}. Therefore, from the decomposition of M above, it is easy to see that ker p_{1} ∩ ker p_{2} ∩ ⋯ ∩ ker p_{k} = (id − p_{1})(id − p_{2}) ⋯ (id − p_{k})M. □

Henceforth, let

e = (id − p_{1})(id − p_{2}) ⋯ (id − p_{k})  (1.6)

so that

eM ≅ 𝔰𝔩_{n}^{⊗k}.  (1.7)

The centralizer algebra End_{𝔤}(M) of transformations commuting with the action of 𝔤 = 𝔤𝔩_{n} on M = V^{⊗k} ⊗ (V^{∗})^{⊗k} was investigated in [BCHLLS], where it was shown to be a homomorphic image of a certain algebra B_{k,k}(n) of diagrams with walls. A diagram in B_{k,k}(n) consists of two rows of vertices with 2k vertices in each row. There is a wall separating the first k vertices on the left in each row from the k vertices on the right. Each vertex is connected to precisely one edge but with the requirement that horizontal edges must cross the wall, but vertical edges cannot cross. The product d_{1}d_{2} of two diagrams d_{1} and d_{2} is obtained by placing d_{1} above d_{2}, identifying the bottom row of d_{1} with the top row of d_{2}, and following the resulting paths. Cycles in the middle are deleted, but there is a scalar factor, which is n to the number of middle cycles. For example, in B_{5,5}(n) we would have the following product,

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The group S_{k} × S_{k} acts on M, where the first copy of the symmetric group S_{k} acts on the first k factors and the second copy on the next k factors by place permutation. These actions commute with the 𝔤-action, and so afford transformations in End_{𝔤}(M). There is representation ϕ : B_{k,k}(n) → End_{𝔤}(M) of the algebra B_{k,k}(n) on M which commutes with the 𝔤-action. Under this representation, the diagrams in B_{k,k}(n) having no horizontal edges are mapped to the place permutations coming from S_{k}×S_{k}. The identity element in B_{k,k}(n) is just the diagram with each node in the top row connected to the one directly below it in the second row, and it maps to the identity transformation in End_{𝔤}(M). Under ϕ, a diagram such as the one pictured below is mapped to a contraction mapping (in this case to c_{3,1}).

• • • • • • • • • •  • • • • • • • • • •^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  (1.8)

It is shown in [BCHLLS] that the algebra End_{𝔤}(M) is generated by S_{k} × S_{k} and the contraction maps c_{i,j}, and the above mapping ϕ is an isomorphism if n ≥ 2k. Moreover [BCHLLS] describes the projection maps onto the irreducible summands of M in the following way.

Suppose for some integer r satisfying 0 ≤ r ≤ k that \underline{s} = \{s_{1}, …, s_{k−r}\} and \underline{t} = \{t_{1}, …, t_{k−r}\} are ordered subsets of \{1, …, k\} of cardinality k − r, and define the following product

c_{\underline{s},\underline{t}} =^{def} c_{s_{1},t_{1}} ⋯ c_{s_{k−r},t_{k−r}}  (1.9)

of the contraction maps c_{s_{i},t_{i}}. Then c_{\underline{s},\underline{t}} belongs to the centralizer algebra End_{𝔤}(M). There is a corresponding product of diagrams in B_{k,k}(n) like the one displayed in (1.8), which ϕ maps onto c_{\underline{s},\underline{t}}.

Assume λ = (λ_{1} ≥ λ_{2} ≥ ⋯) is a partition of r. Associated to λ is its Young frame or Ferrers diagram having λ_{i} boxes in the ith row. A standard tableau is a filling of the boxes in the diagram of λ in such a way that the entries increase from left to right across each row and down each column. Let T be a standard tableau of shape λ with entries in \underline{s}^{c} = \{1, …, k\} \ \{s_{1}, …, s_{k−r}\}. Associated to T is its Young symmetrizer

y_{T} = ( ∑_{ρ∈R_{T}} ρ)( ∑_{γ∈C_{T}} sgn(γ)γ),  (1.10)

where the first sum ranges over the row group of T, which consists of all permutations in S_{k} that transform each entry of T to an entry in the same row, and the second sum is over the column group of T of permutations that move each entry of T to an entry in the same column. For example,

y_{1 5 4} = (id + (1 5))(id − (1 4)),

which belongs to the group algebra ℂS_{k} of the symmetric group S_{k}. The map y_{T} is an essential idempotent, that is, there is an integer m so that y_{T}^{2} = my_{T}.

Similarly, assume for some partition μ ⊢ r that T^{∗} is a standard tableau of shape μ with entries chosen from \underline{t}^{c} = \{1, …, k\}\\{t_{1}, …, t_{k−r}\}. The mapping

y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}}  (1.11)

is an essential idempotent in End_{𝔤}(M). (Note that here we are supposing that y_{T} acts on the factors in V^{⊗k} and y_{T^{∗}} on the factors in (V^{∗})^{⊗k} by place permutations, and that id is the identity map on V^{⊗k} or (V^{∗})^{⊗k}, respectively.) Moreover, y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}}M is isomorphic to the irreducible 𝔤-module L(λ, μ) having highest weight given by the partitions λ and μ. The collection of all maps y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} as r = 0, 1, …, k; \underline{s}, \underline{t} range over all possible choices of ordered subsets of cardinality k − r in {1, …, k}; λ and μ range over all partitions of r; and T (resp. T^{∗}) ranges over all standard tableaux of shape λ (resp. μ) with entries in \underline{s}^{c} (resp. in \underline{t}^{c}), give all the projections onto the irreducible summands of M (this can be found in [BCHLLS]).

Now for the idempotent e in (1.6) we may apply the standard result,

End_{𝔤}(𝔰𝔩_{n}^{⊗k}) ≅ End_{𝔤}(eM) = eEnd_{𝔤}(M)e |_{eM},  (1.12)

(see for example, [CR, Lemma 26.7] or [BBL, Prop. 1.1]).

Lemma 1.13. Assume y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}}. If c_{\underline{s},\underline{t}} contains one of the contraction maps c_{j,j} for some j = 1, …, k, then ey = 0 = ye.

Proof. The mappings y_{T} , y_{T^{∗}}, c_{s_{i},t_{i}}, i = 1, …, k − r, all commute with one another as they operate on different tensor factors. If one of the contraction maps in y equals c_{j,j} = np_{j}, then moving it to the far right produces a product p_{j}e = p_{j}(id − p_{j}) ∏_{ℓ≠j}(id − p_{ℓ}) = 0 in ye, so ye = 0. The argument for ey is similar. □

In [BCHLLS, Def. 2.4] (compare also [H1]) a certain simple tensor x_{T,T^{∗},\underline{s},\underline{t}} = u_{1} ⊗ ⋯ ⊗ u_{k} ⊗ w_{1}^{∗} ⊗ ⋯ w_{k}^{∗} of M is constructed via the algorithm

u_{p} = \{ \begin{matrix}v_{1} & if p ∈ \underline{s} \\ v_{j} & if p ∈ \underline{s}^{c} and p is in the jth row of T \end{matrix}
w_{p}^{∗} = \{ \begin{matrix}v_{1}^{∗} & if p ∈ \underline{t} \\ v_{n−j+1}^{∗} & if p ∈ \underline{t}^{c} and p is in the jth row of T^{∗} \end{matrix}

(1.14)

When y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} is applied to the simple tensor x = x_{T,T^{∗},\underline{s},\underline{t}} the result yx is a nonzero highest weight vector for yM. Moreover, all the highest weight vectors in M are produced in this fashion.

Observe that the factors in x lie in {v_{1}, …, v_{r}, v_{1}^{∗}, v_{n}^{∗}, …, v_{n+1−r}^{∗}}. When the pair (s_{i}, t_{i}) belongs to (\underline{s}, \underline{t}), then the vector v_{1} lies in slot s_{i} in V^{⊗k}, and v_{1}^{∗} lies in slot t_{i} in (V^{∗})^{⊗k}. Replace v_{1} by v_{r+i} and v_{1}^{∗} by v_{r+i}^{∗} in slots s_{i} and t_{i} for i = 1, …, k − r, to produce a new simple tensor x′. Then yx = yx′, as the effect of applying a contraction to v_{1} ⊗ v_{1}^{∗} or to v_{r+i} ⊗ v_{r+i}^{∗} is the same. However, if s_{i} ≠ t_{i} for any i = 1, …, k − r, then p_{j}x′ = 0 for all j. The reason for this is that the vector factors of x′ are a subset of {v_{1}, …, v_{r}, v_{r+1}, …, v_{k}, v_{n}^{∗}, …, v_{n+1−r}^{∗}, v_{k}^{∗}, …, v_{r+1}^{∗}}. If n ≥ 2k, these are all distinct. As s_{i} ≠ t_{i} for any i = 1, …, k − r, slot j on the left and slot j on the right do not contain a pair of dual vectors (of the form v_{ℓ}, v_{ℓ}^{∗}). Therefore p_{j}x′ = 0 for all j and ex′ = x′.

These calculations show that yeM ≠ 0, as it contains yex′ = yx′ = yx, which is a maximal vector of weight (λ, μ). But then yeM ⊆ yM, and the irreducibility of yM forces yeM = yM. To summarize we have

Proposition 1.15. Assume n ≥ 2k. If y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} and s_{i} ≠ t_{i} for any pair (s_{i}, t_{i}) in (\underline{s}, \underline{t}), then yeM = yM, an irreducible 𝔤-module of highest weight (λ, μ), where λ is the shape of T and μ is the shape of T^{∗}.

Let c_{j} denote the diagram in B_{k,k}(n) corresponding to the contraction c_{j,j}, but scaled by a factor of 1/n. Then under the representation ϕ : B_{k,k}(n) → End_{𝔤}(M), c_{j} is sent to p_{j}, and b = ∏_{j=1}^{k}(1 − c_{j}) is mapped to the idempotent e.

Let us consider the subspace A spanned by the diagrams d having no forbidden pairs. By a forbidden pair we mean that the ith node on the left is connected to the ith node on the right of the wall either in the top or in the bottom row of d for some i = 1, …, k.

We claim that the map B_{k,k}(n) → bB_{k,k}(n)b is injective on the subspace A of diagrams with no forbidden pairs. Indeed, ∑_{d∈A} a_{d}d ↦ ∑_{d∈A} a_{d}bdb = ∑_{d∈A} a_{d}d + f, where a_{d} ∈ ℂ and f is a linear combination of diagrams in B_{k,k}(n) having at least one forbidden pair. The reason for this is that when diagrams are multiplied, the horizontal edges in the top row of the top diagram and the horizontal edges in the bottom row of the bottom diagram always appear in the resulting product diagram.

Assume y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} is such that s_{i} ≠ t_{i} for any i = 1, …, k − r, and consider the map yM = yeM → eyeM given by yem ↦ eyem. We have argued that yeM = yM, an irreducible 𝔤-module. Therefore, this map either is an injection or is identically zero. In the latter case, eye must be the zero transformation in End_{𝔤}(M). But there is a linear combination z of diagrams in A which maps to y under the representation ϕ, and bzb ↦ eye. Because the product bzb is nonzero, and the representation ϕ is faithful on M if n ≥ 2k, eye must be nonzero. Therefore yM = yeM → eye M is an injection. But it is clearly surjective and a 𝔤-module map, so eye M ≅ yM, an irreducible 𝔤-module with highest weight (λ, μ). We have proved part (1) of the following:

Theorem 1.16. Assume n ≥ 2k, 𝔤 = 𝔤𝔩_{n}, and M = V^{⊗k} ⊗ (V^{∗})^{⊗k}.

(1) Let y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}}, where s_{i} ≠ t_{i} for any i = 1, …, k − r. Then eye(eM) = eyeM is an irreducible 𝔤-submodule of eM of highest weight (λ, μ) where λ is the shape of T and μ is the shape of T^{∗}.

(2) 𝔰𝔩_{n}^{⊗k} ≅ eM = ⨁_{y} eyeM, where the sum is over all y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} such that s_{i} ≠ t_{i} for any i.

Proof. What remains to be shown is that eM = ⨁_{y} eyeM. Observe that because

M = ⨁_{T,T^{∗},\underline{s},\underline{t}} y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}}M,  (1.17)

([BCHLLS, Thm. 2.11], compare also [H2]),

eM = ∑_{y} eyM = ∑_{y} eyeM,  (1.18)

where the sum is over all y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} such that s_{i} ≠ t_{i} for any i. We need to argue that the decomposition eM = ∑_{y}(eye)eM (over such y) is direct.

We have shown previously that yM = yeM and the map E : yeM → eyeM given by yem ↦ eyem is an isomorphism of 𝔤-modules for y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} such that s_{i} ≠ t_{i} for any i. Fix one such idempotent y′ and consider the intersection

ey′eM ∩ ∑_{y≠y′} eyeM.

of ey′eM with the sum over the remaining ones. Then

ey′eM ∩ ∑_{y≠y′} eyeM →^{E^{−1}} y′eM ∩ ∑_{y≠y′} yeM = y′M ∩ ∑_{y≠y′} yM

But y′M ∩ ∑_{y≠y′} yM = 0 by (1.17). Thus, the sum in (1.18) is direct and we have (2). □

§2. Multiplicities

Knowing that

𝔰𝔩_{n}^{⊗k} ≅ eM = ⨁_{y} eyeM,

where the sum is over all y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} such that s_{i} ≠ t_{i} for any i, we may deduce the multiplicity of a particular irreducible summand in 𝔰𝔩_{n}^{⊗k} labelled by (λ, μ), where λ, μ ⊢ r and r = 0, 1, …, k. That multiplicity is the number of y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} with T having shape λ, T^{∗} having shape μ, and c_{\underline{s},\underline{t}} having no pairs s_{i} = t_{i}. Counting the number of c_{\underline{s},\underline{t}} with at least j factors of the form

c_{ℓ,ℓ}, we have \binom{k}{j} for the choice of those contractions, \binom{k − j}{k − r − j} choices

for the remaining s_{i}’s in \underline{s}, and \binom{k − j}{k − r − j} for the rest of the t_{i}’s in \underline{t}, and

(k − r − j)! for the number of ways to pair the chosen s_{i}’s with the chosen t_{i}’s. Thus, the number of such y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} with at least j contractions of the form c_{ℓ,ℓ} is

\binom{k}{j}\binom{k − j}{k − r − j}^{2}(k − r − j)!f^{λ}f^{μ} = \binom{k}{j}\binom{k − j}{r}^{2}(k − r − j)!f^{λ}f^{μ}, (2.1)

where f^{λ} (resp. f^{μ}) is the number of standard tableaux of shape λ, (resp. μ). Therefore, by the inclusion-exclusion principle, we have the following result.

Theorem 2.2. When n ≥ 2k, the multiplicity m_{λ,μ}^{k} in 𝔰𝔩_{n}^{⊗k} of the irreducible 𝔤 = 𝔤𝔩_{n}-module L(λ, μ) with highest weight (λ, μ), where λ, μ ⊢ r, is

m_{λ,μ}^{k} = f^{λ}f^{μ}( ∑_{j=0}^{k−r}(−1)^{j}\binom{k}{j}\binom{k − j}{r}^{2}(k − r − j)!).  (2.3)

For a partition λ of r, the number f^{λ} of standard tableaux of shape λ is given by the well-known hook length formula

f^{λ} = \frac{r!}{h(λ)},

where h(λ) = ∏_{(i,j)∈λ} h_{i,j}, the product of the hook lengths of the boxes of λ. Thus, h_{i,j} is the number of boxes in the (i, j) hook of λ: the number of boxes to the right of (i, j) plus the number of boxes below (i, j) plus 1.

As a result, the expression for the multiplicity of the summand labelled by (λ, μ) also can be written as

m_{λ,μ}^{k} = \frac{1}{h(λ)h(μ)} ∑_{j=0}^{k−r}(−1)^{j} \frac{k!(k − j)!}{j!(k − r − j)!}.  (2.4)

Let us consider a few interesting special cases. The multiplicity of the trivial 𝔤-module in 𝔰𝔩_{n}^{⊗k} (that is, the dimension of the space of 𝔤-invariants) is

m_{∅,∅}^{k} = ∑_{j=0}^{k}(−1)^{j}\binom{k}{j}(k − j)! = k! ∑_{j=0}^{k}(−1)^{j} \frac{1}{j!} = D_{k},  (2.5)

which is the number of derangements on the set {1, …, k} (permutations with no fixed elements). For small values of k, this number is given by

\begin{matrix}k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ D_{k} & 0 & 1 & 2 & 9 & 44 & 265 & 1854 & 14, 833 \end{matrix}  (2.6)

Next, we compute the number of times the adjoint module 𝔰𝔩_{n} = L( , ) occurs in 𝔰𝔩_{n}^{⊗k}. Using the fact that 𝔰𝔩_{n} is self-dual as a 𝔤-module, we see that the number of times 𝔰𝔩_{n} appears in 𝔰𝔩_{n}^{⊗k} is the number of times the trivial module appears in 𝔰𝔩_{n}^{⊗k} ⊗ 𝔰𝔩_{n} = 𝔰𝔩_{n}^{⊗(k+1)}. Hence, the number of times 𝔰𝔩_{n} appears in 𝔰𝔩_{n}^{⊗k} is

m_{,}^{k} = D_{k+1}.  (2.7)

This can also be derived from (2.4) which gives

\begin{aligned}m_{,}^{k} & = ∑_{j=0}^{k−1}(−1)^{j} \frac{k!(k − j)}{j!} = ∑_{j=0}^{k}(−1)^{j} \frac{k!(k − j)}{j!} \\ & = k ∑_{j=0}^{k}(−1)^{j} \frac{k!}{j!} + ∑_{j=1}^{k}(−1)^{j−1} \frac{k!}{(j − 1)!} \\ & = k ∑_{j=0}^{k}(−1)^{j} \frac{k!}{j!} + k ∑_{j=0}^{k−1}(−1)^{j} \frac{(k − 1)!}{j!} \\ & = k(D_{k} + D_{k−1}) = D_{k+1}. \end{aligned}  (2.8)

The last equality in (2.8) is a linear recurrence relation satisfied by the derangement numbers (see for example, [B, (6.5)]).

For any 𝔤-module X,

X ⊗ X^{∗} ≅ End(X)

where the action on the right is (g · ψ)(x) = gψ(x) − ψ(gx) for all g ∈ 𝔤, ψ ∈ End(X), and x ∈ X. Considering the 𝔤-invariants on both sides, we see that

(X ⊗ X^{∗})^{𝔤} ≅ End(X)^{𝔤} = End_{𝔤}(X).  (2.9)

DERANGEMENTS AND TENSOR POWERS OF 𝔰𝔩_{n} 11

Now applying this to X = 𝔰𝔩_{n}^{⊗k} ≅ X^{∗}, we have

End_{𝔤}(𝔰𝔩_{n}^{⊗k}) ≅ (𝔰𝔩_{n}^{⊗2k})^{𝔤}  (2.10)

Consequently,

dim End_{𝔤}(𝔰𝔩_{n}^{⊗k}) = m_{∅,∅}^{2k} = D_{2k},  (2.11)

the number of derangements on a set of 2k elements.

We conclude by displaying the multiplicities m_{λ,μ}^{k} for k = 4. By double centralizer theory, it follows that

dim End_{𝔤}(𝔰𝔩_{n}^{⊗k}) = ∑_{λ,μ⊢r≤k} (m_{λ,μ}^{k})^{2}.

The reader can verify that the squares of the numbers in the following tables do indeed sum to D_{8} = 14, 833. Example: m_{λ,μ}^{4}:

1  3  2  3 1
3  9  6  9 3
2  6  4  6 2
3  9  6  9 6
1  3  2  3 1

12  24 12 24  48 24 12  24 12  42 42 42 42  44  \begin{aligned}& ∅ \\ ∅ & 9 \end{aligned}

§3. The Centralizer Algebra

Now we consider the centralizer algebra C = End_{𝔤}(𝔰𝔩_{n}^{⊗k}) = End_{𝔰𝔩_{n}}(𝔰𝔩_{n}^{⊗k}) and its representation theory. As has already been pointed out in (1.12), we have an isomorphism

C ≅ eEnd_{𝔤}(M)e  (3.1)

where e is the idempotent defined in (1.6). We also have a representation ϕ : B_{k,k}(n) → End(M) which commutes with the 𝔤-action on M. Thus the image of this representation lies in the commuting algebra End_{𝔤}(M). In [BCHLLS, Thm. 5.8] it was shown that ϕ induces an algebra isomorphism

B_{k,k}(n) ≅ End_{𝔤}(M)  (3.2)

for n ≥ 2k. Let b ∈ B_{k,k}(n) be given as above, b = ∏_{j=1}^{k}(1 − c_{j}) where c_{j} is the diagram corresponding to the contraction map c_{j,j} but scaled by 1/n. Then ϕ maps b onto e, and we obtain the following

Proposition 3.3. Let n ≥ 2k. The map ϕ induces an algebra isomorphism between bB_{k,k}(n)b and C = End_{𝔤}(𝔰𝔩_{n}^{⊗k}). Moreover, the set of all elements of the form bdb, as d ranges over all diagrams with no forbidden pairs, is a basis for bB_{k,k}(n)b.

Proof. The first claim follows from the remarks above, so only the second claim remains to be proved. We observe that left (resp., right) multiplication by b kills any diagram with a forbidden pair in its top (resp., bottom) row. Since the diagrams form a basis for B_{k,k}(n), the result follows. □

The basis statement of Proposition 3.3 provides another proof of (2.11), that the dimension of the centralizer algebra C is D_{2k}. Indeed, the diagrams with no forbidden pairs are easily seen to be in bijective correspondence with the permutations σ on the set {1, …, 2k} such that σ(i) ≠ i for all i = 1, …, 2k. This correspondence is given by performing two “flips”, which take a walled Brauer diagram to the diagram obtained by first interchanging the rightmost k dots in its top and bottom rows and then switching corresponding dots on the two sides of the wall on the top row while retaining the edges.

Let r ≤ k and let λ, μ be fixed partitions of r. In [BCHLLS] M_{λ,μ} was defined to be the space spanned by all maximal vectors yx, where y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} and x = x_{T,T^{∗},\underline{s},\underline{t}} (notation of (1.14)), for all pairs \underline{s} = \{s_{1}, …, s_{k−r}\}, \underline{t} = {t_{1}, …, t_{k−r}} of ordered subsets of {1, …, k}, and all standard tableaux T (resp., T^{∗}) of shape λ (resp., μ) with entries from \underline{s}^{c} (resp., \underline{t}^{c}). Moreover, for n ≥ 2k, the M_{λ,μ} provide a complete set of pairwise nonisomorphic irreducible modules for the algebra End_{𝔤}(M) (and hence also for B_{k,k}(n)).

Lemma 3.4. Assume n ≥ 2k and let y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}}, x = x_{T,T^{∗},\underline{s},\underline{t}}. Then ey ≠ 0 if and only if s_{i} ≠ t_{i} for all pairs (s_{i}, t_{i}) in (\underline{s}, \underline{t}). Hence eM_{λ,μ} ≠ 0 precisely when this condition can be satisfied, and in that case, eM_{λ,μ} is the linear span of all the nonzero eyx, y and x as above.

Proof. This follows from results in [BCHLLS], Lemma 1.13, and its converse, which is in the paragraph before Theorem 1.16. □

We remark that for y, x as in the preceding lemma, we have eyx = eyx′ = eyex′, where x′ is the modified simple tensor described in Section 1.

Moreover, it is easy to see that eM_{λ,μ} = 0 when λ = μ = ∅ and k = 1, for in that case it is impossible to construct a y = y_{T} y_{T^{∗}}c_{\underline{s},\underline{t}} satisfying the condition s_{i} ≠ t_{i} for all pairs (s_{i}, t_{i}) in (\underline{s}, \underline{t}). In all other cases eM_{λ,μ} ≠ 0 when n ≥ 2k.

Theorem 3.5. Assume n ≥ 2k. The collection of all nonzero eM_{λ,μ} for λ, μ partitions of r, r = 0, 1, …, k, forms a complete set of pairwise nonisomorphic irreducible modules for the algebra C ≅ bB_{k,k}(n)b.

Proof. It is well-known that if u is an idempotent in an algebra A, the functor u(−) (sometimes called the Schur functor; see [G, 6.2]) taking A-modules to uAu-modules is an exact covariant functor which maps an irreducible module to either an irreducible module or zero. In the particular case that A = B_{k,k}(n) and u = b, this functor takes the irreducible module M_{λ,μ} to bM_{λ,μ} = eM_{λ,μ}. □

Theorem 3.6. Assume n ≥ 2k. Then as a bimodule for C × 𝔤,

𝔰𝔩_{n}^{⊗k} ≅ eM ≅ ⨁_{r=0}^{k} ⨁_{λ,μ⊢r} eM_{λ,μ} ⊗ L(λ, μ),

where the decomposition is into pairwise nonisomorphic irreducible modules for C × 𝔤.

Proof. This follows from the previous results and standard double-centralizer theory. □

For n ≥ 2k the dimension of the irreducible C-module eM_{λ,μ} is given by m_{λ,μ}^{k} (see Theorem 2.2).

References

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[BCHLLS] G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, C. Lee, and J. Stroomer, Tensor product representations of general linear groups and their connections with Brauer algebras, J. Algebra 166 (1994), 529-567.

[B] R.A. Brualdi, Introductory Combinatorics, 3rd ed., Prentice Hall, Englewood Cliffs, N.J., 1999.

[C] C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, vol. XI, Pure and Applied Math, Interscience Publ. John Wiley, New York, 1962.

[G] J.A. Green, Polynomial Representations of GL_{n}, Lecture Notes in Math., vol. 830, Springer-Verlag, Heidelberg, 1980.

[H1] P. Hanlon, On the construction of the maximal vectors in the tensor algebra of 𝔤𝔩_{n}, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence R.I., 1984, pp. 73–80.

[H2] P. Hanlon, On the decomposition of the tensor algebra of the classical Lie algebras, Adv. in Math. 56 (1985), 238–282.

Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 benkart@math.wisc.edu

Department of Mathematical and Computer Sciences, Loyola University Chicago, Chicago, Illinois 60626 doty@math.luc.edu