A Note on Cayley-Hamilton Theorem for Generalized Matrix Function

M. R. Darafsheh, K. Mallahi, M. R. Pournaki

Abstract Let $_{n} be the symmetric group on n letters and G be a subgroup of $_{n}. Suppose χ is an irreducible complex character of G and d_{χ}^{G} is the generalized matrix function afforded by G and χ. In this paper we characterize for which G and χ, the Cayley-Hamilton Theorem holds.

Keywords: Generalized matrix function, Hilbert-Nullstellensatz. 1995 Mathematics Subject Classification: Primary 15A69; Secondary 20C30.

1 Introduction

Let $_{n} be the symmetric group on n letters, and G be a subgroup of $_{n}. Suppose c : G → ℂ is a complex valued function on G and let M_{n}(ℂ) denote the set of all n by n matrices over ℂ. The function d_{c}^{G} : M_{n}(ℂ) → ℂ with definition

d_{c}^{G}(A) = ∑_{σ∈G} c(σ) ∏_{t=1}^{n} a_{tσ(t)}

where A = (a_{ij}), 1 ≤ i, j ≤ n, is called the Schur function afforded by G and c, in particular if c = χ : G → ℂ, is an irreducible complex character of G, then d_{χ}^{G} is called the generalized matrix function (see [3]). This function is introduced by I. Schur in [4].

Note that if c = ϵ : $_{n} → ℂ is the alternating character of $_{n}, then d_{ϵ}^{$_{n}} is the determinant function, i.e., d_{ϵ}^{$_{n}} = det. If c = 1 : $_{n} → ℂ is the identity character 1, then d_{1}^{$_{n}} is the permanent function, i.e., d_{1}^{$_{n}} = per. In particular for an irreducible complex character χ of $_{n}, d_{χ}^{$_{n}} is called the immanant function. For more information about the complex irreducible characters of the symmetric group we refer the reader to [1].

Many familiar properties which hold for the determinant do not hold for d_{c}^{G}. For example as stated in [5] if A and R are n by n matrices with R invertible, then d_{c}^{G}(A) = d_{c}^{G}(R^{−1}AR) does not necessarily hold. But it is proved in [2] that if R is a permutation matrix, then d_{c}^{G}(A) = d_{c}^{G}(R^{−1}AR).

Definition 1 Suppose G ≤ $_{n} and c : G → ℂ is a complex valued function on G and A ∈ M_{n}(ℂ). The characteristic polynomial of A afforded by the Schur function d_{c}^{G} is denoted by f_{A}^{G,c}(x) and is defined by f_{A}^{G,c}(x) = d_{c}^{G}(xI_{n} − A).

In the case of G = $_{n} and χ the irreducible complex character of G, the polynomial f_{A}^{G,χ}(x) is called the χ-th immanantal polynomial of A (see [3]). This polynomial has been studied in variety of aspects. We see from the definition, that the characteristic polynomial of A afforded by d_{c}^{G}, f_{A}^{G,c}(x), is a polynomial over ℂ with degree n, i.e., f_{A}^{G,c}(x) ∈ ℂ[x]. Note that if G = $_{n} and c = ϵ the alternating character of $_{n}, then f_{A}^{$_{n},ϵ}(x) is the ordinary characteristic polynomial of A which is defined in Linear Algebra.

A property of the determinant is that the classical Cayley-Hamilton Theorem holds for it. The following example shows that the Cayley-Hamilton Theorem does not hold for d_{c}^{G} in general.

Example 2 Let G = {(1), (12)} < $_{3} and c = ϵ, the alternating character of G, and

A = (\begin{matrix}1 & 1 & 1 \\ 2 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}) ∈ M_{3}(ℂ).

Then the characteristic polynomial of A afforded by d_{c}^{G} is

f_{A}^{G,c}(x) = d_{c}^{G}(xI_{3} − A) = d_{c}^{G} (\begin{matrix}x − 1 & −1 & −1 \\ −2 & x − 1 & −1 \\ 0 & 0 & x − 1 \end{matrix}) = x^{3} − 3x^{2} + x + 1,

and note that A does not satisfy f_{A}^{G,c}(x).

In this paper we prove that if d_{c}^{G} has the property of the Cayley-Hamilton Theorem, then G must be the full symmetric group and c must be a multiple of the alternating character of G, in particular if the generalized matrix function d_{χ}^{G} has the property of the Cayley-Hamilton Theorem, then G must be the full symmetric group and χ must be the alternating character of G, i.e., the Cayley-Hamilton Theorem holds only for the determinant function, in the family of generalized matrix functions. Roughly speaking the Cayley-Hamilton Theorem is true only for determinants!

2 Main Theorem

Definition 3 Suppose G ≤ $_{n} and c : G → ℂ is a complex valued function on G. We say d_{c}^{G} has the Cayley-Hamilton property if for every A ∈ M_{n}(ℂ) we have f_{A}^{G,c}(A) = 0.

Note that if G = $_{n} and χ = ϵ the alternating character of $_{n}, then by the classical Cayley-Hamilton Theorem d_{ϵ}^{$_{n}} = det has the Cayley-Hamilton property.

Lemma 4 Let c : $_{n} → ℂ be a complex valued function on $_{n}. If d_{c}^{$_{n}} has the Cayley-Hamilton property, then for any singular matrix A ∈ M_{n}(ℂ), we have d_{c}^{$_{n}}(A) = 0.

Proof. Suppose f_{A}^{$_{n},c}(x) ∈ ℂ[x] is of the form

f_{A}^{$_{n},c}(x) = a_{n}x^{n} + ⋯ + a_{1}x + a_{0}.

Suppose d_{c}^{$_{n}}(A) ≠ 0. Since f_{A}^{$_{n},c}(0) = a_{0} = d_{c}^{$_{n}}(−A) = (−1)^{n}d_{c}^{$_{n}}(A), so we obtain a_{0} ≠ 0. By hypotheses d_{c}^{$_{n}} has the Cayley-Hamilton property and therefore f_{A}^{$_{n},c}(A) = 0, i.e., a_{n}A^{n} +⋯+a_{1}A+a_{0}I = 0 and so A.\frac{−1}{a_{0}}(a_{n}A^{n−1} +⋯+a_{1}I) = I, and hence A is invertible which is a contradiction. □

Let c : $_{n} → ℂ be a complex valued function on $_{n}. If X = (x_{ij}) is an n by n matrix whose entries are n^{2} indeterminates x_{ij} over ℂ, then d_{c}^{$_{n}}(X) and in particular det(X), is a polynomial over n^{2} indeterminates x_{11}, x_{12}, …, x_{nn}. In this way, we have the following Lemma. Note that this Lemma is stated in [3] (Lemma 8.3, page 267) without proof and here we provide a proof for it.

Lemma 5 det(X) is an irreducible polynomial over ℂ[x_{11}, x_{12}, …, x_{nn}].

Proof. If det(X) is reducible, then we have det = P.Q, where P and Q are nonconstant polynomials in ℂ[x_{11}, x_{12}, …, x_{nn}]. If x_{ij} appears in P for some i and j, then since det is linear in each variable the polynomial Q cannot contain terms involving x_{ik} or x_{kj}, 1 ≤ k ≤ n, which are in the same column or row as x_{ij}. So Q is constant, and this is a contradiction. □

Lemma 6 Let c : $_{n} → ℂ be a complex valued function on $_{n}. If d_{c}^{$_{n}} has the Cayley-Hamilton property, then det(X) | d_{c}^{$_{n}}(X) in ℂ[x_{11}, x_{12}, …, x_{nn}], where X = (x_{ij}) is an arbitrary n by n matrix over ℂ.

Proof. Consider I_{1} = ⟨d_{c}^{$_{n}}(X)⟩ and I_{2} = ⟨det(X)⟩, the ideals generated by d_{c}^{$_{n}}(X) and det(X) respectively in ℂ[x_{11}, x_{12}, …, x_{nn}]. Suppose for the ideal I of ℂ[x_{11}, x_{12}, …, x_{nn}], V (I) denotes the affine ℂ-variety in ℂ^{n^{2}},

V (I) = {(a_{11}, a_{12}, …, a_{nn}) ∈ ℂ^{n^{2}}| f(a_{11}, a_{12}, …, a_{nn}) = 0, for all f ∈ I},

and for the subset Y of ℂ^{n^{2}}, J(Y) denotes the set

J(Y) = {f ∈ ℂ[x_{11}, x_{12}, …, x_{nn}]| f(a_{11}, a_{12}, …, a_{nn}) = 0, for all (a_{11}, a_{12}, …, a_{nn}) ∈ Y }.

By Lemma 4, we have V (I_{2}) ⊆ V (I_{1}) and therefore J(V (I_{1})) ⊆ J(V (I_{2})) so by Hilbert-Nullstellensatz, we obtain \sqrt{I_{1}} ⊆ \sqrt{I_{2}}. Since I_{1} ⊆ \sqrt{I_{1}}, so I_{1} ⊆ \sqrt{I_{2}}. Because d_{c}^{$_{n}}(X) ∈ I_{1}, we have d_{c}^{$_{n}}(X) ∈ \sqrt{I_{2}} and this means there exists k ∈ ℕ such that (d_{c}^{$_{n}}(X))^{k} ∈ I_{2} and so det(X) | (d_{c}^{$_{n}}(X))^{k}. But by Lemma 5, det(X) is irreducible over ℂ[x_{11}, x_{12}, …, x_{nn}] and therefore det(X)|d_{c}^{$_{n}}(X) in ℂ[x_{11}, x_{12}, …, x_{nn}]. □

Corollary 7 Let c : $_{n} → ℂ be a complex valued function on $_{n}. If d_{c}^{$_{n}} has the Cayley-Hamilton property, then there is a constant k ∈ ℂ, such that c = kϵ, where ϵ is the alternating character of $_{n}.

Proof. By Lemma 6, det(X)|d_{c}^{$_{n}}(X) in ℂ[x_{11}, x_{12}, …, x_{nn}], but det(X) and d_{c}^{$_{n}}(X) are of degree one in each indeterminate, therefore d_{c}^{$_{n}}(X) = k det(X), where k ∈ ℂ is a constant, and so c = kϵ. □

Corollary 8 Let G ≨ $_{n} and c : G → ℂ be a nonzero complex valued function on G. Then d_{c}^{G} does not have the Cayley-Hamilton property.

Proof. Assume d_{c}^{G} has the Cayley-Hamilton property. Define the function c′ : $_{n} → ℂ by

c′(σ) = \{ \begin{matrix}c(σ) & , & if σ ∈ G, \\ 0 & , & if σ ∈ $_{n} \ G. \end{matrix}

Then for all A = (a_{ij}) ∈ M_{n}(ℂ),

d_{c}^{G}(A) = ∑_{σ∈G} c(σ) ∏_{t=1}^{n} a_{tσ(t)} = ∑_{σ∈$_{n}} c′(σ) ∏_{t=1}^{n} a_{tσ(t)} = d_{c′}^{$_{n}}(A),

therefore d_{c}^{G} = d_{c′}^{$_{n}}. Because d_{c}^{G} has the Cayley-Hamilton property, d_{c′}^{$_{n}} satisfies this property too and hence by Corollary 7, c′ = kϵ, where k ∈ ℂ is a constant. By definition of c′ we obtain k = 0, so c′ = 0 and therefore c = 0. This contradiction shows d_{c}^{G} does not have Cayley-Hamilton property. □

Main Theorem Let G ≤ $_{n} and χ be an irreducible complex character of G. Then d_{χ}^{G} has the Cayley-Hamilton property if and only if G = $_{n} and χ = ϵ, the alternating character of $_{n}, i.e., d_{χ}^{G} = det.

Proof. If d_{χ}^{G} has the Cayley-Hamilton property, then by Corollary 8, G = $_{n}. So d_{χ}^{G} = d_{χ}^{$_{n}} and by Corollary 7, we have χ = kϵ, where k ∈ ℂ is a constant. But χ is an irreducible character of G = $_{n} and therefore we have k = 1, so χ = ϵ, hence d_{χ}^{G} = d_{ϵ}^{$_{n}} = det. The converse of the Theorem holds by the classical Cayley-Hamilton Theorem. □

References

[1] G. James, A. Kerber, “The Representation Theory of the Symmetric Group”, Addision-Wesley, Reading, MA, 1981.

[2] D. E. Littlewood, “The Theory of Group Characters”, Oxford University Press, London, 1958.

[3] R. Merris, “Multilinear Algebra”, Gordon and Breach Science Publishers, 1997.

[4] I. Schur, Über Endliche Gruppen und Hermitesche Formen, Math. Z. 1 (1918), 184– 207.

[5] J. Turner, Generalized Matrix Functions and the Graph Isomorphism Problem, SIAM J. Appl. Math. 16 (1968), 520–526.

The Authors’ Addresses

M. R. Darafsheh, Department of Mathematics and Computer Science, University of Tehran, and Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran.

E-mail address: darafshe@vax.ipm.ac.ir

K. Mallahi, Department of Mathematical Sciences, Sharif University of Technology, and Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran.

E-mail address: keivanmk@vax.ipm.ac.ir

M. R. Pournaki, Department of Mathematics and Computer Science, University of Tehran, and Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran.

E-mail address: pournaki@vax.ipm.ac.ir