Deformation of Cayley's hyperdeterminants
aa r X i v : . [ m a t h . QA ] J un Deformation of Cayley’s hyperdeterminants
Tommy Wuxing Cai and Naihuan Jing
Abstract.
We introduce a deformation of Cayley’s second hyperde-terminant for even-dimensional hypermatrices. As an application, weformulate a generalization of the Jacobi-Trudi formula for Macdonaldfunctions of rectangular shapes generalizing Matsumoto’s formula forJack functions.
1. Introduction
For a 2 m dimensional hypermatrix A = ( A ( i , · · · , i m )) ≤ i ,...,i m ≤ n Cayley’s second hyperdeterminant [ ] of A is a generalization of the usualdeterminant:det [2 m ] ( A ) = 1 n ! X σ ,...,σ m ∈ S n m Y i =1 sgn( σ i ) n Y i =1 A ( σ ( i ) , . . . , σ m ( i )) . (1.1)In [ ] properties of hyperdeterminants and discriminants are studied ingeneral. In particular, it is easy to see that det [2 m ] ( A ) is invariant when A is replaced by B ◦ k A , where the action is defined for example, ( B ◦ A )( i , · · · , i m ) = P nj =1 B i ,j A ( j, i , · · · , i m ) for any B ∈ GL n . This meansthat the hyperdeterminant is an invariant under the action of GL ⊗ mn . Ingeneral, it is a challenging problem to come up with relative invariants underthe (non-diagonal) action of GL n × · · · × GL n .A q -analog Hankel determinant has been studied in [ ] (see also [ ])and q -analog of the hyperdeterminant for non-commuting matrices has beenintroduced in the context of quantum groups [ ].In this paper, we introduce a λ -hyperdeterminant (of commuting en-tries) by replacing permutations with alternating sign matrices in a natu-ral and nontrivial manner. We will show that the new hyperdeterminantdeforms Cayley’s hyperdeterminant. As an application, we give a Jacobi-Trudi type formula for Macdonald polynomials to generalize the previous Mathematics Subject Classification.
Primary: 05E05; Secondary: 17B69, 05E10.
Key words and phrases.
Hyperdeterminants, λ -determinants, Macdonaldpolynomials.*Corresponding author: Naihuan Jing. formula of Matsumoto for Jack polynomials of rectangular shapes. It wouldbe interesting to uncover the relation with q -Hankel determinants. Also therelationship with q -deformations remains mysterious. λ -hyperdeterminants To introduce the λ -hyperdeterminant, we first consider generalization ofthe permutation sign in the usual determinant, and then generalize Cayley’shyperdeterminants.A permutation matrix is any square matrix obtained by permuting therows (or columns) of the identity matrix, therefore its determinant is equalto the sign of the corresponding permutation. Each row or column of a fixedpermutation matrix has only one nonzero entry along each row and each col-umn. Generalizing the notion of permutation matrices, an alternating signmatrix is a square matrix with entries 0’s, 1’s and ( − × Q = − . Clearly the nonzero entries in each row or column of an alternating signmatrix must start with 1. We denote by Alt n the set of n by n alternatingsign matrices and P n the subset of permutation matrices of size n . The set P n , endowed with the usual matrix multiplication, is a group isomorphicto the symmetric group S n . As n increases, Alt n contains more and morenon-permutation matrices. In fact, it is known [
17, 10 ] that Alt n has acardinality of Q n − i =0 (3 i +1)!( n + i )! .For X = ( x ij ) ∈ Alt n , we define the inversion number of X by [ ] i ( X ) = X r>i,s
Let λ be a parameter and A an n × n matrix. Following Robbins andRumsey [ ], we introduce the λ -determinant as follows.det λ ( A ) = X X ∈ Alt n ( − λ ) i ( X ) (1 − λ − ) n ( X ) A X = X π ∈ S n ( − λ ) i ( π ) a π · · · a nπ n + X X ∈ Alt n \ P n ( − λ ) i ( X ) (1 − λ − ) n ( X ) A X . (2.2)Note that the second equality holds even for λ = 1 due to 0 = 1. Weremark that the original λ -determinant [ ] corresponds to a sign change ofours, while our definition deforms the usual determinant at λ = 1. In factwhen λ = 1, the second summand vanishes, therefore det ( A ) reduces to theusual determinant. In general, det λ ( A ) is a rational function in the variable a ij or rather a polynomial function in the variable a ± ij . For any 3 × A = ( a ij )det λ ( A ) = a a a − λa a a − λ a a a − λa a a + λ a a a + λ a a a + λ (1 − λ ) a a a a a , where the last term is due to the alternating sign matrix Q in (2.1).Another example is a λ -deformation of the Vandermonde determinant[ ]: For commutating variables x i , 1 ≤ i ≤ n , one has thatdet λ ( x j − i ) = Y ≤ i 1, this formula has an interesting connection with thePfaffian Pf: det ( x j − i )det − ( x j − i ) = Y ≤ i 1, then there are odd number ofnonzero entries along the i th row and they are alternatively ± j , · · · , j k +1 th columns such that 1 ≤ j < j < · · · < j k +1 ≤ n and k ≥ 1. Therefore X ( i ) = n X j =1 jx ij = j − j + j − j + · · · − j k + j k +1 = j + ( j − j ) + · · · + ( j k +1 − j k ) ≥ k + 1and n X j =1 jx ij = ( j − j ) + · · · + ( j k − − j k ) + j k +1 < j k +1 ≤ n. Hence one always has that 1 ≤ X ( i ) = P nj =1 jx ij ≤ n for any alternatingsign matrix X = ( x ij ).For fixed real numbers q and a , we use the conventional q -analogue of a :(2.7) ( a ; q ) n = (1 − a )(1 − aq ) · · · (1 − aq n − ) . With this preparation, we now introduce the notion of λ -hyperdeterminants. Definition . Let M = ( M ( i , i , . . . , i m )) ≤ i ,i ,...,i m ≤ n be a 2 m dimensional hypermatrix, the λ -hyperdeterminant of M is defined bydet [2 m ] λ ( M ) = (1 − λ ) n ( λ ; λ ) n X X ,X ,...,X m ∈ Alt n φ λ ( X , X , . . . , X m )(2.8) × n Y i =1 M ( X ( i ) , X ( i ) , . . . , X m ( i )) , where the generalized sign factor φ λ ( X , . . . , X m ) is the following expression m Y r =1 ( − λ r − ) i ( X r ) (1 − λ r − ) n ( X r ) ( − λ r ) i ( X m + r ) (1 − λ r ) n ( X m + r ) , (2.9)and the generalized permutation ( X (1) , . . . , X ( n )) for X ∈ Alt n was definedin (2.6). EFORMATION OF CAYLEY’S HYPERDETERMINANTS 5 Note that the λ -hyperdeterminant is different from the q -hyperdeterminantdefined on quantum linear group [ ].We also remark that alternating sign matrices with n ( X ) > X is therefore alwaystaken as a permutation matrix. Moreover, one can even fix the first indicesto be 1 , , . . . , n and drop the quantum factor.We now explain how the λ -hyperdeterminant deforms the Cayley hyper-determinant. Proposition . For any m -dimensional hypermatrix A = ( a i ,...,i m ) ≤ i ,...,i m ≤ n one has that (2.10) lim λ → det [2 m ] λ ( A ) = det [2 m ] ( A ) . Proof. When λ = 1 the summands in the right side of (2.8) (omittingthe factor) vanish if one of X i ’s has at least one negative entry. Thus onlypermutation matrices contribute to the sum, so we can let X r = P σ r =( δ jσ r ( i ) ) ∈ S n for r = 1 , · · · , m . Subsequently X r ( i ) = P nj =1 jX r ( i, j ) = P nj =1 jδ jσ r ( i ) = σ r ( i ), and we see that the right side of (2.8) matches exactlywith that of (1.1). (cid:3) 3. Hyperdeterminant formula for Macdonald functions We now generalize Matsumoto’s hyperdeterminant formula for Jack poly-nomials [ ] as an application of the λ -hyperdeterminant.Recall that the classical Jacobi-Trudi formula expresses the Schur func-tion associated to partition µ as a determinant of simpler Schur functionsof row shapes: s µ ( x ) = det( s µ i − i + j ( x )) . (3.1)In [ ], Matsumoto gave a simple formula for Jack functions of rectan-gular shapes using the hyperdeterminant (see also [ ]). Proposition . [ ] Let k, s, m be positive integers. The rectangularJack functions Q ( k s ) ( m − ) can be expressed compactly as follows. ( sm )!( m !) s Q ( k s ) ( m − ) = s !det [2 m ] ( Q k + P mj =1 ( i m + j − i j ) ) ≤ i ,...,i m ≤ s . (3.2)As the Schur function s µ is the specialization Q µ (1) of the Jack func-tion, one sees immediately that this formula specializes to the Jacobi-Trudiformula for rectangular Schur functions.Macdonald symmetric functions Q µ ( q, t ) [ ] are a family of orthogonalsymmetric functions with two parameters q, t and labeled by partition µ . Inthis paper we consider the case that t = q m , m any positive integer. When q approaches to 1, Q µ ( q, q m ) specializes to the Jack symmetric function Q µ ( m − ) [ ] which has many applications (eg. [ ]). When the partition TOMMY WUXING CAI AND NAIHUAN JING µ = ( k s ), the symmetric function Q µ ( q, t ) or Q µ ( m − ) is referred as therectangular (shaped) Macdonald function.We now state our main result. Theorem . Let k, s, m be positive integers. Up to a scalar factor,the rectangular Macdonald function Q ( k s ) ( q, q m ) is a m -dimensional λ -hyperdeterminant: ( q ; q ) sm ( q ; q ) sm Q ( k s ) ( q, q m )= ( q ; q ) s (1 − q ) s det [2 m ] q ( Q k + P mr =1 ( i m + r − i r ) ( q ; q m )) ≤ i ,...,i m ≤ s , (3.3) where det [2 m ] q is the λ -hyperdeterminant with λ = q . Remark . When q = 1, one recovers the hyperdeterminant formula(3.2) for Jack functions of rectangular shapes. The formula also gives an-other expression of the general formula of Lassalle-Schlosser [ ] (see also[ ]) in the special shape.To prepare its proof, we first recall the following result from [ ]. Proposition . Let ρ = ( k s ) be an rectangular partition with k, s > . Let m be a positive integer. Then ( q ; q ) sm ( q ; q ) sm Q ρ ( q, q m ) is the coefficient of z k z k · · · z ks in the Laurent polynomial F G , where F = Y ≤ i Recall that the q -Dyson Laurent polynomial (3.4) is a gener-alization of the Vandemonde polynomial. Using (2.3), we can rewrite the EFORMATION OF CAYLEY’S HYPERDETERMINANTS 7 q -Dyson Laurent polynomial in the z i using λ -determinants. F = Y ≤ i NJ gratefully acknowledges the partial support of Simons Foundationgrant no. 523868, Humbolt Foundation, NSFC grant no. 11531004 andMPI for Mathematics in the Sciences in Leipzig during this work. References [1] H. Belbachir, A. Boussicault, J.-G. Luque, Hankel hyperdeterminants, rectangularJack polynomial and even powers of the Vandermonde , J. Algebra 320 (2008), 3911–3925.[2] W. Cai, N. Jing, On vertex operator realizations of Jack functions , J. Alg. Combin.32 (2010), 579–595.[3] T. W. Cai, N. Jing, Jack vertex operators and realization of Jack functions , J. Alg.Combin. 39 (2014), 53–74.[4] T. W. Cai, Macdonald symmetric functions of rectangular shapes , J. Combin. Th. A128 (2014), 162–179.[5] A. Cayley, On the theory of determinants , Collected Papers, vol. 1, Cambridge Univ.Press, Cambridge, 1889, pp. 63–80.[6] F. J. 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