A simple counterexample for the permanent-on-top conjecture
aa r X i v : . [ m a t h . C O ] J a n A simple counterexample for the permanent-on-topconjecture
Tran Hoang AnhE¨otv¨os Lor´and University. Institute of Mathematics.E-mail: [email protected]: E. Frenkel P´eterNovember 06, 2020
Abstract
The permanent-on-top conjecture (POT) was an important conjecture on the largesteigenvalue of the Schur power matrix of a positive semi-definite Hermitian matrix, for-mulated by Soules. The conjecture claimed that for any positive semi-definite Hermitianmatrix H , per ( H ) is the largest eigenvalue of the Schur power matrix of the matrix H .After half a century, the POT conjecture has been proven false by the existence of coun-terexamples which are checked with the help of computer. It raises concerns about acounterexample that can be checked by hand (without the need of computers). A newsimple counterexample for the permanent-on-top conjecture is presented which is a com-plex matrix of dimension 5 and rank 2. The symbol S n denotes the symmetric group on n objects. The permanent of a square matrixis a vital function in linear algebra that is similar to the determinant. For an n × n matrix A = ( a ij ) with complex coefficients, its permanent is defined as per ( A ) = P σ ∈ S n Q ni =1 a i,σ ( i ) .By H n we mean the set of all n × n positive semi-definite Hermitian matrices. The Schur powermatrix of a given n × n matrix A = ( a ij ), denoted by π ( A ), is a n ! × n ! matrix with the elementsindexed by permutations σ, τ ∈ S n : π στ ( A ) = n Y i =1 a σ ( i ) τ ( i ) Conjecture 1. The permanent-on-top conjecture(POT) [1]:
Let H be an n × n positivesemi-definite Hermitian matrix, then per ( H ) is the largest eigenvalue of π ( H ).In 2016, Shchesnovich provided a 5-square, rank 2 counterexample to the permanent-on-topconjecture with the help of computer [2]. Definition 1.1.
For an n × n matrix A = ( a ij ), let d A be a function S n → C defined by d A ( σ ) = n Y i =1 a σ ( i ) i This function is also called the ”diagonal product” function [3]. Then we can define det ( A ) = P σ ∈ S n ( − sign ( σ ) d A ( σ ) and per ( A ) = P σ ∈ S n d A ( σ ).1or any n -square matrix A and I, J ⊂ [ n ], A [ I, J ] denotes the submatrix of A consisting ofentries which are the intersections of i -th rows and j -th columns where i ∈ I, j ∈ J . We define A ( I, J ) = A [ I c , J c ].In this paper, we shall study the properties of the spectrum of the Schur power matrix byexamining the spectra of the matrices C k ( A ) which are defined in the manner:For any 1 ≤ k ≤ n , the matrix C k ( A ) is a matrix of size (cid:0) nk (cid:1) × (cid:0) nk (cid:1) with its ( I, J ) entry ( I and J are k -element subsets of [ n ]) defined by per ( A [ I, J ]) .per ( A [ I c , J c ]). There is another conjectureon these matrices C k ( A ) which states that: Conjecture 2. Pate’s conjecture [4]
Let A be an n × n positive semi-definite Hermitianmatrix and k be a positive integer number less than n , then the largest eigenvalue of C k ( A ) is per ( A ).Pate’s conjecture is weaker than the permanent-on-top conjecture(POT) because it is well-known that every eigenvalue of C k is also an eigenvalue of the Schur power matrix. In the case k = 1, in [3], it was conjectured that per ( A ) is necessarily the largest eigenvalue of C ( A ) if A ∈ H n . Stephen W. Drury has provided an 8-square matrix as a counterexample for this casein the paper [5]. Besides, Bapat and Sunder raise a question as follows: Conjecture 3. Bapat & Sunder conjecture:
Let A and B = ( b ij ) be n × n positive semi-definite Hermitian matrices, then per ( A ◦ B ) ≤ per ( A ) n Y i =1 b ii where A ◦ B is the entrywise product (Hadamard product).The Bapat & Sunder conjecture is weaker than the permanent-on-top conjecture and hasbeen proved false by a counterexample which is a positive semi-definite Hermitian matrix oforder 7 proposed by Drury [6]. In the present paper, a new simple counterexample for thepermanent-on-top conjecture and Pate’s conjecture is presented. It has size 5 × Conjecture 4. The Lieb permanent dominance conjecture 1966 [10]
Let H be a sub-group of the symmetric group S n and let χ be a character of degree m of H . Then1 m X σ ∈ H χ ( σ ) n Y i =1 a iσ ( i ) ≤ per ( A )holds for all n × n positive semi-definite Hermitian matrix A.The permanent dominance conjecture is weaker than the permanent-on-top conjecture andstill open. The POT conjecture was proposed by Soules in 1966 as a strategy to prove thepermanent dominance conjecture. Definition 1.2.
The elementary symmetric polynomials in n variables x , x , . . . , x n are e k for k = 0 , , . . . , n . In this paper, we define e k ( x i ) for i = 1 , , . . . , n to be the elementarysymmetric polynomial of degree k in n − x i from theset { x , x , . . . , x n } . We define the associated matrix of a matrix representation W : S n → GL N ( C ) with respect toa n × n matrix A by: M W ( A ) = X σ ∈ S n d A ( σ ) W ( σ )2 roposition 2.1. The Schur power matrix of a given n × n Hermitian matrix A is the associatedmatrix of the left-regular representation with respect to A .Proof. Take a look at the ( σ, τ ) entry of M L ( A ) which is X η ∈ S n , η ◦ τ = σ d A ( η ) = d A ( σ ◦ τ − ) = n Y i =1 a σ ( i ) τ ( i ) the right side is the ( σ, τ ) entry of π ( A ). (cid:4) Let us now consider two important matrices C ( A ) amd C ( A ) that shall appear frequentlyfrom now on. Definition 2.1.
Let N k : S n → GL ( nk )( C ) be the matrix representation given by the permuta-tion action of S n on (cid:0) [ n ] k (cid:1) . Proposition 2.2.
For any n × n Hermitian matrix A , the matrix C k ( A ) is the matrix M N k ( A ) . We obtain directly the statement that every eigenvalue of matrix M N k ( A ) is an eigenvalueof the associated matrix of the left-regular representation which is the Shur power matrix. Con-sequently, Pate’s conjecture is weaker than the permanent-on-top conjecture(POT). C ( A ) in rank 2 case The main object of this section is n × n positive semi-definite Hermitian matrices of rank 2.We know that every matrix A ∈ H n of rank 2 can be written as the sum v v ∗ + v v ∗ where v and v are two column vectors of order n . Definition 3.1.
A matrix A ∈ H n is called ”formalizable” if A can be written in the form v v ∗ + v v ∗ and every element of v vector is non-zero. Definition 3.2.
The formalized matrix A ′ of a given formalizable matrix A defined in themanner: if A = v v ∗ + v v ∗ and v = ( a , . . . , a n ) T , a i = 0 ∀ i = 1 , . . . , n ; v = ( b , . . . , b n ) T then A ′ = v v ∗ + v v ∗ where v = (1 , . . . , T and v = ( b a , . . . , b n a n ) T . Proposition 3.1. : Let A ∈ H n be a formalizable matrix, then π ( A ) = Q ni =1 k a i k π ( A ′ ) .Proof. We compare the ( σ, τ )-th entries of two matrices. π στ ( A ) = n Y i =1 ( a σ ( i ) a τ ( i ) + b σ ( i ) b τ ( i ) ) = n Y i =1 k a i k n Y i =1 b σ ( i ) a σ ( i ) b τ ( i ) a τ ( i ) ! = n Y i =1 k a i k π στ ( A ′ ) (cid:4) Remark.
The same result will be obtained with the matrices C k ( A ) and C k ( A ′ ). It is obviousto see that if the matrix A is a counterexample for the permanent-on-top conjecture and Pate’sconjecture then so is A ′ . Assume that we have an unformalizable matrix B ∈ H of rank 2that is a counterexample for the permanent-on-top conjecture and Pate’s conjecture. That alsoimplies that there is a column vector x such that the following inequality holds x ∗ π ( B ) x k x k > per ( B )3y continuity and B = vv ∗ + uu ∗ , we can change slightly the zero elements of the vector v suchthe the inequality remains. Therefore, if the permanent-on-top conjecture or Pate’s conjectureis false for some positive semi-definite Hermitian matrix of rank 2 then so is the permanent-on-top conjecture and Pate’s conjecture for some formalizable matrices. That draws our attentionto the set of all formalizable matrices.For any n × n positive semi-definite Hermitian matrix A of rank 2 there exist two eigenvectorsof v and u of A such that A = vv ∗ + uu ∗ . Let u i , v i be the i -th row elements of v and u respectively for i = 1 , n . In the case A has a zero row then per ( A ) = 0 and the Schur powermatrix and matrices C k ( A ) of A are all zero matrices, there is nothing to discuss. Otherwise,every row of A has a non-zero element (so does every column since A is a Hermitian matrix)which means that for any i = 1 , n , the inequalities k v i k + k u i k > A can berewritten in the form(sin( x ) v + cos( x ) u )(sin( x ) v + cos( x ) u ) ∗ + (cos( x ) v − sin( x ) u )(cos( x ) v − sin( x ) u ) ∗ ∀ x ∈ [0 , π ]and the system of n equations sin( x ) v i + cos ( x ) u i = 0 , i = 1 , n takes finite solutions in theinterval [0 , π ]. Therefore, there exists x ∈ [0 , π ] satisfying that (sin( x ) v + cos ( x ) u ) has everyelement different from 0. Hence, every rank 2 positive semi-definite Hermitian that has nozero-row is formalizable. Several properties about the formalized matrices are presented below.Let H ∈ H n be a formalizable matrix of the form H = vv ∗ + uu ∗ where v = (1 , . . . , T and u = ( x , x , . . . , x n ) T . We recall quickly the Kronecker product [8]. Definition 3.3.
The Kronecker product (also known as tensor product or direct product) oftwo matrices A and B of sizes m × n and s × t , respectively, is defined to be the ( ms ) × ( nt )matrix A ⊗ B = a B a B . . . a n Ba B a B . . . a n B ... ... ... a n B a n B . . . a nn B Lemma 3.1.
The upper bound of rank of the Schur power matrix of rank 2: If A is n × n of rank 2 then rank of π ( A ) is not larger than n − n .Proof. We observe that rank ( A ) = 2 implies that dim ( Im ( A )) = 2 and dim ( Ker ( A )) = n − h w, t i be an orthonormal basis of the orthogonal complement of Ker ( A ) in C n , then denote v = Aw, u = At . Thus, A can be rewritten in the form vw ∗ + ut ∗ where v = ( a , . . . , a n ) T , u =( b , . . . , b n ) T . It is obvious that Im ( A ) = h v, u i . Let us denote the Kronecker product of n copies of the matrix A by ⊗ n A . The mixed-product property of Kronecker product implies that Im ( ⊗ n A ) = h{⊗ ni =1 t i , t i ∈ { v, u }}i . Furthermore, the Schur power matrix of A is a diagonalsubmatrix of ⊗ n A obtained by deleting all entries of ⊗ n A that are products of entries of A having two entries in the same row or column. Let define the function f in the manner that f : {⊗ ni =1 t i , t i ∈ { v, u }} → e V and the σ -th element of f ( ⊗ ni =1 α i ) vector of order n ! is Q ni =1 t i ( σ ( i )) where t i ( j ) is the j -th rowelement of the column vector t i . Let B = { f ( ⊗ ni =1 t i ) , t i ∈ { v, u }} then B is a generator of Im ( π ( A )) since π ( A ) is a principal matrix of ⊗ n A and Im ( ⊗ n A ) = h{⊗ ni =1 t i , t i ∈ { v, u }}i . Wepartition B into disjoint sets S k k = 0 , , . . . , n, S k = { f ( ⊗ ni =1 t i ) , t i ∈ { v, u } , v appears k times in the Kronecker product } Hence, for any k = 1 , , . . . , n the σ -th row element of the sum vector P w ∈ S k w is X ≤ i <...
The permanent of a formalized matrix [9]: per ( H ) = n X k =0 k !( n − k )! k e k k Proof.
We show that per ( H ) = X σ ∈ S n n Y i =1 (1 + x i x σ ( i ) )= n ! + X σ ∈ S n n X k =1 X ≤ i <...
The matrix C ( H ) can be rewritten in the form C ( H ) = per ( H ) n vv ∗ + n − X k =1 ( k − n − − k )! n v k v ∗ k where v = (1 , . . . , T of order n , for k = 1 , . . . , n − , v k = ( . . . , ne k ( x i ) − ( n − k ) e k | {z } i-th element , . . . ) T Proposition 3.3.
For any k = 1 , . . . , n − , h v, v k i = 0 Proof. h v, v k i = n X i =1 ( ne k ( x i ) − ( n − k ) e k )= n n X i =1 e k ( x i ) − n ( n − k ) e k = 0 (cid:4) Proposition 3.4.
The rank of C ( H ) is the cardinality of the set { x i , i = 1 , n } . In formula, rank ( C ( H )) = |{ x i , i = 1 , n }| .Proof. For the i -th element of v k , we have ne k ( x i ) − ( n − k ) e k = ke k − nx i e k − ( x i )= ke k + n k X j =1 ( − j e k − j x ji which lead us to a conclusion that h v, v , . . . , v n − i = h p , . . . , p n − i where p j = ( . . . , x ji |{z} i-th element , . . . ) T which is equal to |{ x i , i = 1 , n }| by the determinantal formula of Vandermonde matrices. (cid:4) roposition 3.5. The determinant of C ( H ) is given by det ( C ( H )) = per ( H ) n n − Y k =1 n ( k − n − − k )! · Y i Case 1: There are indices i and j such that x i = x j then rank ( C ( H )) < n that isequivalent to det ( C ( H )) = 0.Case 2: x i ’s are distinct then { v, v , . . . , v n − } makes a basis of C n . Therefore, C ( H ) is similarto the Gramian matrix of n vectors (cid:26)q per ( H ) n v ; q ( k − n − − k )! n v k , k = 1 , n − (cid:27) . Thus det ( C ( H )) = det G r per ( H ) n v ; r ( k − n − − k )! n v k , k = 1 , n − !! = per ( H ) n n − Y k =1 ( k − n − − k )! n · det ( G ( v, v , . . . , v n − ))And from the proof of proposition 3.4, we obtain that( v, v , . . . , v n − ) = ( p , p , . . . , p n − ) . . . ke k . . . ( n − e n − . . . ( − ne k − . . . ( − ne n − . . . . .. . . . .. . . . . . . . . . . ( − j ne k − j . . .. . . . .. . . . .. . . . . . . . . . . . . . ( − n − n The matrix in the right side is the transition matrix given byThe ( i, j )-th entry = ( − i ne j − i if i > j − e j − if i = 1 and j > 11 if ( i, j ) = (1 , e = 1; e t = 0 if t < 0. Moreover, we observe that the transition matrixis an upper triangular matrix with the absolute value of diagonal entries equal to n except the(1 , p , p , . . . , p n − ) is a Vandermonde matrix. Hence det ( C ( H )) = per ( H ) n n − Y k =1 n ( k − n − − k )! · det ( G ( p , p , . . . , p n − ))= per ( H ) n n − Y k =1 n ( k − n − − k )! · k det ( p , p , . . . , p n − ) k = per ( H ) n n − Y k =1 n ( k − n − − k )! · Y i Let H = vv ∗ + uu ∗ be an n × n positive semi-definite Hermitian matrix then: det ( C ( H )) = per ( H ) n n − Y k =1 n ( k − n − − k )! · Y i 120 + 24 c , c, c , c , c } Moreover, the spectrum of π ( H ) is • per ( H ) = 120 + 24 c of multiplicity 1 • c, c , c , c of multiplicity 4 8 c , c of multiplicity 5 • c = 2 is a solution of the inequality 120 + 24 c − c < 0. Therefore, thematrix H = vv ∗ + 2 uu ∗ where v = (1 , . . . , T , u = ( i, − , − i, , T is a counterexample to thepermanent-on-top conjecture(POT). H = − i − i 11 + 2 i − i − − i − i − i − i To continue, we prove the matrix H is a counterexample for Pate’s conjecture in the case n = 5and k = 2. For the purposes of this paper let us describe the tensor product of vector spacesin terms of bases: Definition 4.1. Let V and W be vector spaces over C with bases { v i } and { w i } , respectively.Then V ⊗ W is the vector space spanned by { v i ⊗ w j } subject to the rules:( αv + α ′ v ′ ) ⊗ w = α ( v ⊗ w ) + α ′ ( v ′ ⊗ w ) v ⊗ ( αw + α ′ w ′ ) = α ( v ⊗ w ) + α ′ ( v ⊗ w ′ )for all v, v ′ ∈ V and w, w ′ ∈ W and all scalars α, α ′ .If h , i is an inner product on V then we can define an inner product h , i on V ⊗ V in the manner: h v i ⊗ v i , v i ⊗ v i i = h v i , v i ih v i , v i i for any v i , v i , v i , v i vectors.On C [ x, y ], we consider the inner product, and the resulting Euclidean norm | · | , such thatmonomials are orthogonal and | x n y k | = n ! k !. Proposition 4.1. The permanent of the Gram matrix of any 1-forms f j ∈ C x ⊕ C y is | Q f j | .Proof. We prove the generalization of the statement which states that if f , f , . . . , f n , g , g , . . . , g n be 2 n A be an n × n matrix with ( i, j )-th entry h f i , g j i , then per ( A ) = * n Y i =1 f i , n Y i =1 g i + Let f i = α i x + β i y, g i = α ′ i x + β ′ i y for any i ∈ { , , . . . , n } .We compute each side of the equality: per ( A ) = X σ ∈ S n n Y i =1 h f i , g σ ( i ) i = X σ ∈ S n n Y i =1 h α i x + β i y, α ′ σ ( i ) x + β ′ σ ( i ) y i = X σ ∈ S n n Y i =1 ( α i · α σ ( i ) + β i · β ′ σ ( i ) )= X σ ∈ S n n X k =0 X ≤ i <...
4) and f = x . Their Gram matrix is the positive definiteHermitian matrix H = − i − i 11 + 2 i − i − − i − i − i − i with perH = | f f f f f | = | x − xy | = 5! + 16 · 4! = 504 (according to the proposition 4.1) . When { p, q, r, s, t } = { , , , , } , define F p,q = f p f q ⊗ f r f s f t and an inner product on C [ x, y ] ⊗ C [ x, y ] as the definition 4.1. It is obvious that C ( H ) of H is the Gram matrix of the ten tensors F p,q with { p, q, r, s, t } = { , , , , } , p < q , and r < s < t . We observe that(1 + i ) F + ( − i ) F + ( − − i ) F + (1 − i ) F − iF + 2 F + 2 iF − F = 16 √ x ⊗ y − √ xy ⊗ xy + 16 √ y ⊗ x y, whose norm squared is 2 · · 2! + 2 · ·