View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 435 (2011) 1085–1098 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa The stabilizer of immanants Ke Ye Texas A&M University, Dept. of Mathematics, College Station, TX 77843-3368, United States ARTICLE INFO ABSTRACT Article history: Immanants are homogeneous polynomials of degree n in n2 vari- Received 6 May 2010 ables associated to the irreducible representations of the symmetric Accepted 15 February 2011 group S of n elements. We describe immanants as trivial S mod- Available online 17 March 2011 n n ules and show that any homogeneous polynomial of degree n on SubmittedbyR.Loewy the space of n × n matrices preserved up to scalar by left and right AMS classification: action by diagonal matrices and conjugation by permutation ma- 15 linear and multilinear algebra trices is a linear combination of immanants. Building on works of 51 geometry Duffner [5] and Purificação [3], we prove that for n ≥ 6 the identity component of the stabilizer of any immanant (except determinant, Keywords: permanent, and π = (4, 1, 1, 1))is(Sn) T(GLn × GLn) Z2, Immanants where T(GLn ×GLn) is the group consisting of pairs of n×n diagonal Stabilizers matrices with the product of determinants 1, acting by left and right Representation theory matrix multiplication, (S ) is the diagonal of S × S , acting by Lie algebra n n n conjugation (Sn is the group of symmetric group) and Z2 acts by sending a matrix to its transpose. Based on the work of Purificação and Duffner [4], we also prove that for n ≥ 5 the stabilizer of the immanant of any non-symmetric partition (except determinant and permanent) is (Sn) T(GLn × GLn) Z2. © 2011 Elsevier Inc. All rights reserved. 1. Introduction Littlewood [7] defined polynomials of degree n in n2 variables generalizing the notion of determi- nant and permanent, called immanants, and are defined as follows. Definition 1.1. For any partition π n, define a polynomial of degree n in matrix variables (xij)n×n associated to π as follows: n Pπ := χπ (σ ) xiσ(i). σ ∈Sn i=1 This polynomial is called the immanant associated to π. E-mail address: [email protected] 0024-3795/$ - see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2011.02.029 1086 K. Ye / Linear Algebra and its Applications 435 (2011) 1085–1098 Example 1.2. If π = (1, 1,...,1) then Pπ is exactly the determinant of the matrix (xij)n×n. π = ( ) π n σ() Example 1.3. If n then P is the permanent σ ∈Sn i=1 xi i . Remark 1.4. Let E and F be Cn. Since immanants are homogeneous polynomials of degree n in n2 variables, we can identify them as elements in Sn(E ⊗ F) (Identify the space E ⊗ F with the space of n×n matrices). The space Sn(E ⊗F) is a representation of GL(E ⊗F), in particular, it is a representation of GL(E) × GL(F) ⊂ GL(E ⊗ F). So we can use the representation theory of GL(E) × GL(F) to study immanants. The explicit expression of an immanant Pπ in Sn(E ⊗ F) is: n χπ (σ ) ei ⊗ fσ(i), σ ∈Sn i=1 where is interpreted as the symmetric tensor product. In Section 3, we remark that immanants can be defined as trivial Sn modules (Proposition 3.3). Duffner found the system of equations determining the stabilizer of immanants (except determinant and permanent) for n ≥ 4in[5] in year 1994. Two years later, Purificação proved in [3] that if the system of equations in [5] has a solution, then permutations τ1 and τ2 in the system must be the same. Building on works of Duffner and Coelho, We prove the main results Theorems 1.5 and 1.7 of this paper in Section 4. Theorem 1.5. Let π be a partition of n ≥ 6 such that π = (1,...,1), (4, 1, 1, 1) or (n), then the identity component of the stabilizer of the immanant Pπ is (Sn) T(GLn × GLn) Z2,whereT(GLn × GLn) is the group consisting of pairs of n × n diagonal matrices with the product of determinants 1,actingbyleft and right matrix multiplication, (Sn) is the diagonal of Sn × Sn, acting by conjugation, (Sn is the group of symmetric group.) and Z2 acts by sending a matrix to its transpose. Remark 1.6. It is well-known that Theorem 1.5 is true for permanent as well, but our proof does not recover this case. Theorem 1.7. Let n ≥ 5 and let π be a partition of n which is not symmetric, that is, π is not equal to its transpose, and π = (1,...,1) or (n), then the stabilizer of the immanant is (Sn) T(GLn × GLn)Z2. Remark 1.8. One can compute directly from the system of equations determined by Duffner for the case n = 4 and π = (2, 2) and see that in this case, Theorem 1.7 fails, since there will be many additional components. For example, ⎛ ⎞ − − ⎜e e ee⎟ ⎜ ⎟ ⎜ ⎟ ⎜1111⎟ C = ⎜ ⎟ ⎜ − − ⎟ ⎝11 1 1⎠ 1 − 1 1 − 1 e e e e stabilizes the immanant P(2,2), but it is not in the identity component. 2. Notations and preliminaries (1) E and F are n-dimensional complex vector spaces. { }n { }n (2) ei i=1 and fi i=1 are fixed basis of E and F, respectively. (3) Sn is the symmetric group on n elements. Given σ ∈ Sn, we can express σ as disjoint product of cycles, and we can denote the conjugacy class of σ by (1i1 2i2 ,...,nin ), meaning that σ is a disjoint product of i1 1-cycles, i2 2-cycles,…, in n-cycles. Sometimes we might use (k1, k2,...,kp) K. Ye / Linear Algebra and its Applications 435 (2011) 1085–1098 1087 ≥ ≥ ≥ ··· ≥ ≥ p = σ (where n k1 k2 kp 1 and i=1 ki n) to indicate the cycle type of .This notation means that σ contains a k1-cycle, a k2-cycle and a kp-cycle. (4) π n is a partition of n, we can write π = (π ,...,π ) with π ≥···≥π . 1 k 1 k (5) π is the conjugate partition of π. (6) Mπ (sometimes we might use [π] as well) is the irreducible representation of the symmetric group Sn corresponding to the partition π. (7) Sπ (E) is the irreducible representation of GL(E) corresponding to π. (8) χπ is the character of Mπ . (9) Let V be a representation of SL(E), then the weight-zero-subspace of V is denoted as V . ⊗ 0 (10) For Sπ (E), we have a realization in the tensor product E n via the young symmetrizer cπ : Sπ (E) ∼ ⊗ = cπ (E n),where π is any young tableau of shape π. (11) The action of Sn on E is the action of the Weyl group of SL(E), that is, the permutation repre- sentation on Cn. (12) Fix a young tableau π,definePπ¯ := {σ ∈ Sn | σ preserves each row of π¯ } and Qπ¯ := {σ ∈ Sn | σ preserves each column of π¯ }. (13) For a polynomial P in variables (xij)n×n, denote the stabilizer of P in GL(E ⊗ F) by G(P). 3. The description of immanants as modules Consider the action of T(E) × T(F) on immanants, where T(E),T(F) are maximal tori (diagonal matrices) of SL(E), SL(F), respectively. For any (A, B) ∈ T(E) × T(F), ⎛ ⎞ ⎛ ⎞ ... ... ⎜a1 0 0 ⎟ ⎜b1 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ... ⎟ ⎜ ... ⎟ ⎜ 0 a2 0 ⎟ ⎜ 0 b2 0 ⎟ = ⎜ ⎟ , = ⎜ ⎟ . A ⎜ . ⎟ B ⎜ . ⎟ ⎜ . .. ⎟ ⎜ . .. ⎟ ⎝ ⎠ ⎝ ⎠ 00... an 00... bn π = χπ (σ ) n σ() ( , ) π For the immanant P σ ∈Sn i=1 xi i , the action of A B on P is given by n (A, B).Pπ := χπ (σ ) aibσ(i)xiσ(i) = Pπ . σ ∈Sn i=1 That is, immanants are in the SL(E) × SL(F) weight-zero-subspace of Sn(E ⊗ F). On the other hand, the decomposition of Sn(E ⊗ F) as GL(E) × GL(F)-modules: Sn(E ⊗ F) = Sλ(E) ⊗ Sλ(F). λn Accordingly, we have a decomposition of the weight-zero-subspace: n (S (E ⊗ F))0 = (Sλ(E))0 ⊗ (Sλ(F))0. λn ∼ Proposition 3.1. For λ n, (Sλ(E))0 = Mλ as Sn-modules. Proof. See [8,p.272]. Thus we can identify (Sλ(E))0 ⊗ (Sλ(F))0 with Mλ ⊗ Mλ as an Sn × Sn module. Also, the diagonal n (Sn) of Sn × Sn is isomorphic to Sn,soMλ ⊗ Mλ is a S -module. Mλ ⊗ Mλ is an irreducible Sn × Sn module, but it is reducible as and Sn-module, so that we can decompose it. Consider the action of Sn on Sn(E ⊗ F),letσ ∈ Sn, then: 1088 K. Ye / Linear Algebra and its Applications 435 (2011) 1085–1098 σ.xij = xσ(i)σ (j). n So immanants are invariant under the action of S , hence are contained in the isotypic component of ( ( )) ⊗ ( ( )) ⊗ the trivial Sn representation of λn Sλ E 0 Sλ F 0 = λn Mλ Mλ. Proposition 3.2. As a Sn module, Mλ ⊗ Mλ contains only one copy of trivial representation. Proof. Denote the character of σ ∈ Sn on Mλ ⊗ Mλ by χ(σ), and let χtrivial be the character of the trivial representation. From the general theory of characters, it suffices to show that the inner product (χ, χtrivial) = 1. First, the character χ(σ,τ)of (σ, τ) ∈ Sn × Sn on the module Mλ ⊗ Mλ is χλ(σ )χλ(τ).