aa r X i v : . [ m a t h - ph ] S e p A note on the Weingarten function
Claudio ProcesiSeptember 18, 2020
Abstract
The aim of this note is to compare work of Formanek [6] on acertain construction of central polynomials with that of Collins [3] onintegration on unitary groups.These two quite disjoint topics share the construction of the samefunction on the symmetric group, which the second author calls
Wein-garten function .By joining these two approaches we succeed in giving a simplifiedand very natural presentation of both Formanek and Collins’s Theory.
We need to recall some basic facts on the representation Theory of thesymmetric and the linear group.Let V be a vector space of finite dimension d over a field F which inthis note can be taken as Q or C . On the tensor power V ⊗ k act both thesymmetric group S k and the linear group GL ( V ), Formula (1.1), furthermoreif F = C and V is equipped with a Hilbert space structure one has an inducedHilbert space structure on V ⊗ k . The unitary group U ( d ) ⊂ GL ( V ) acts on V ⊗ k by unitary matrices. σ · u ⊗ u ⊗ . . . ⊗ u k := u σ − (1) ⊗ u σ − (2) ⊗ . . . ⊗ u σ − ( k ) ,g · u ⊗ u ⊗ . . . ⊗ u k := gu ⊗ gu ⊗ . . . ⊗ gu k , σ ∈ S k , g ∈ GL ( V ) . (1)The first step of Schur Weyl duality is the fact that the two operator algebrasΣ k ( V ) , B k,d generated respectively by S k and GL ( V ) acting on V ⊗ k , areboth semisimple and each the centralizer of the other.In particular the algebra Σ k ( V ) ⊂ End ( V ⊗ k ) = End ( V ) ⊗ k equals thesubalgebra Σ k ( V ) = (cid:0) End ( V ) ⊗ k (cid:1) GL ( V ) of invariants under the conjugationaction of the group GL ( V ) → End ( V ) ⊗ k , g g ⊗ g ⊗ . . . ⊗ g. From this, the double centralizer Theorem and work of Frobenius andYoung one has that, under the action of these two commuting groups, V ⊗ k decomposes into the direct sum V ⊗ k = ⊕ λ ⊢ k, ht ( λ ) ≤ d M λ ⊗ S λ ( V ) (2)1ver all partitions λ of k of height ≤ d , (the height ht ( λ ) denotes the numberof elements, nonzero, of λ ). M λ is an irreducible representation of S k while S λ ( V ), called a Schurfunctor is an irreducible polynomial representation of GL ( V ), which remainsirreducible also when restricted to U ( d ).The character theory of the two groups can be deduced from these rep-resentations. We shall denote by χ λ ( σ ) the character of the permutation σ on M λ . As for S λ ( V ) its character is expressed by a symmetric function S λ ( x , . . . , x d ) restriction to the first d variables of a stable symmetric func-tion called Schur function . Of this deep and beautiful Theory, see [13], [7],[8], [21], [16], we shall use only two remarkable formulas, the hook formula due to Frame, Robinson and Thrall [15], expressing the dimension of χ λ (1)of M λ and the hook-content formula of Stanley, cf. [19, Corollary 7.21.4])expressing the dimension of s λ ( d ) := S λ (1 , . . . ,
1) = S λ (1 d ) of S λ ( V ).We display partitions by Young diagrams, as in (1.3), by ˜ λ we denotethe dual partition obtained by exchanging rows and columns. The boxes , cf.(1.15), of the diagram are indexed by pairs ( i, j ) of coordinates. Given thenone of the boxes u we define its hook number h u and its content c u as follows: Definition . Let λ be a partition of n and let u = ( i, j ) ∈ λ be a box inthe corresponding Young diagram. The hook number h u = h ( i, j ) and the content c u are defined as follows: h u = h ( i, j ) = λ i + ˇ λ j − i − j + 1 , c u = c ( i, j ) := j − i. (3) Example 1.3.
Note that the box u = (3 ,
4) defines a hook in the diagram λ , and h u equals the length (number of boxes) of this hook:43In this figure, we have λ = (13 , , , , ) , ht ( λ ) = 7 with u = (3 , λ = (7 , , , , ) and h u = λ + ˇ λ − − − λ = (8 , , , h u , respectively content c u is written inside its box in thediagram λ : 21 9 7 5 4 3 2 15 3 13 11 , 0 1 2 3 4 5 6 7-1 0 1-2 -1-3 Theorem 1.4 (The hook and hook–content formulas) . Let λ ⊢ k be a parti-tion of k and χ λ (1) and s λ ( d ) the dimension of the corresponding irreduciblerepresentation M λ of S k and S λ ( V ) of GL ( V ) , dim( V ) = d . Then s λ ( d ) = Y u ∈ λ d + c u h u , χ λ (1) = k ! Q u ∈ λ h u . (4)The remarkable Formula of Stanley exhibits s λ ( d ) as a polynomial ofdegree k = | λ | in d with zeroes the integers − c u and leading coefficient Q u ∈ λ h − u . The dual of the algebra
End ( V ) ⊗ k can be identified, in a GL ( V ) equivariantway, to End ( V ) ⊗ k by the pairing formula: h A ⊗ A · · · ⊗ A k | B ⊗ B · · · ⊗ B k i := tr ( A ⊗ A · · · ⊗ A k ◦ B ⊗ B · · · ⊗ B k )= tr ( A B ⊗ A B · · · ⊗ A k B k ) = k Y i =1 tr ( A i B i ) . Under this isomorphism the multilinear invariants of matrices are iden-tified with the GL ( V ) invariants of End ( V ) ⊗ m which in turn are spannedby the elements of the symmetric group, hence by the elements of Formula(5). These are explicited by Formula (6) as in Kostant [11] Proposition 1.5.
The space T d ( k ) of multilinear invariants of k , d × d matrices is identified with End GL ( V ) ( V ⊗ k ) and it is linearly spanned by thefunctions: T σ ( X , X , . . . , X d ) := tr ( σ − ◦ X ⊗ X ⊗ · · · ⊗ X d ) , σ ∈ S k . (5) If σ = ( i i . . . i h ) . . . ( j j . . . j ℓ )( s s . . . s t ) is the cycle decomposition of σ then we have that T σ ( X , X , . . . , X d ) equals = tr ( X i X i . . . X i h ) . . . tr ( X j X j . . . X j ℓ ) tr ( X s X s . . . X s t ) . (6) Proof.
Since the identity of Formula (6) is multilinear it is enough to proveit on the decomposable tensors of
End ( V ) = V ⊗ V ∗ which are the endo-morphisms of rank 1, u ⊗ ϕ : v
7→ h ϕ | v i u .So given X i := u i ⊗ ϕ i and an element σ ∈ S k in the symmetric groupwe have σ − ◦ u ⊗ ϕ ⊗ u ⊗ ϕ ⊗ . . . ⊗ u k ⊗ ϕ k ( v ⊗ v ⊗ . . . ⊗ v k )3 = k Y i =1 h ϕ i | v i i u σ (1) ⊗ u σ (2) ⊗ . . . ⊗ u σ ( k ) u ⊗ ϕ ⊗ u ⊗ ϕ ⊗ . . . ⊗ u k ⊗ ϕ m ◦ σ − ( v ⊗ v ⊗ . . . ⊗ v k )= m Y i =1 h ϕ i | v σ ( i ) i u ⊗ u ⊗ . . . ⊗ u k = k Y i =1 h ϕ σ − ( i ) | v i i u ⊗ u ⊗ . . . ⊗ u k = ⇒ σ − ◦ u ⊗ ϕ ⊗ u ⊗ ϕ ⊗ . . . ⊗ u m ⊗ ϕ k = u σ (1) ⊗ ϕ ⊗ u σ (2) ⊗ ϕ ⊗ . . . ⊗ u σ ( k ) ⊗ ϕ k = ⇒ u ⊗ ϕ ⊗ u ⊗ ϕ ⊗ . . . ⊗ u k ⊗ ϕ k ◦ σ = u ⊗ ϕ σ (1) ⊗ u ⊗ ϕ σ (2) ⊗ . . . ⊗ u k ⊗ ϕ σ ( k ) . (7)So we need to understand in matrix formulas the invariants tr ( σ − u ⊗ ϕ ⊗ u ⊗ ϕ ⊗ . . . ⊗ u k ⊗ ϕ k ) = k Y i =1 h ϕ i | u σ ( i ) i . (8)We need to use the rules u ⊗ ϕ ◦ v ⊗ ψ = u ⊗ h ϕ | v i ψ, tr ( u ⊗ ϕ ) = h ϕ | u i from which the formula easily follows by induction. Remark . We can extend the Formula (5) to the group algebra t ( X τ ∈ S d a τ τ )( X , . . . , X d ) := X τ ∈ S d a τ T τ ( X , X , . . . , X d ) . (9) The algebra of the symmetric group S k decomposes into the direct sum F [ S k ] = ⊕ λ ⊢ k End ( M λ )of the matrix algebras associated to the irreducible representations M λ ofpartitions λ ⊢ k . Denote by χ λ the corresponding character of S k and by e λ ∈ End ( M λ ) ⊂ F [ S k ] the corresponding central unit. These elements forma basis of orthogonal idempotents of the center of F [ S k ].For a finite group G let e be the central idempotent of an irreduciblerepresentation with character χ . One has the Formula: e = χ (1) | G | X g ∈ G χ ( g − ) g. (10)As for the algebra Σ k ( V ), it is isomorphic to F [ S k ] if and only if d ≥ k .Otherwise it is a homomorphic image of F [ S k ] with kernel the ideal generatedby any antisymmetrizer in d + 1 elements. This ideal is the direct sum of the End ( M λ ) with ht ( λ ) > d , where ht ( λ ), the height of λ , cf. page 2 is also thelength of its first column. So thatΣ k ( V ) = ⊕ λ ⊢ k, ht ( λ ) ≤ d End ( M λ ) (11)4 .8 The function Wg ( d, µ ) We start with a computation of a character.
Definition . Given a permutation ρ ∈ S k we denote by c ( ρ ) the numberof cycles into which it decomposes, and π ( ρ ) ⊢ k the partition of k given bythe lengths of these cycles. Notice that c ( ρ ) = ht ( π ( ρ )).Given a partition µ ⊢ k we denote by( µ ) := { ρ | π ( ρ ) = µ } , C µ := X ρ | π ( ρ )= µ ρ = X ρ ∈ ( µ ) ρ. (12)The sets ( µ ) := { ρ | π ( ρ ) = µ } are the conjugacy classes of S k and, think-ing of F [ S k ] as functions from S k to F we have that C µ is the characteristicfunction of the corresponding conjugacy class. Of course the elements C µ form a basis of the center of the group algebra F [ S k ]. Proposition 1.10.
1) For every pair of integers k, d the function P on S k given by P : ρ d c ( ρ ) is the character of the permutation action of S k on V ⊗ k , dim F ( V ) = d .2) The symmetric bilinear form on F [ S k ] given by h σ | τ i := d c ( στ ) hasas kernel the ideal generated by the antisymmetrizer on d + 1 elements. Inparticular if k ≤ d it is non degenerate.Proof.
1) If e , . . . , e d is a given basis of V we have the induced basisof V ⊗ k , e i ⊗ . . . ⊗ e i k which is permuted by the symmetric group. For apermutation representation the trace of an element σ equals the number ofthe elements of the basis fixed by σ .If σ = (1 , , . . . , k ) is one cycle then e i ⊗ . . . ⊗ e i k is fixed by σ if andonly if i = i = . . . = i k are equal, so equal to some e j so tr ( σ ) = d .For a product of a cycles of lengths b , b , . . . b a which up to conjugacywe may consider as(1 , , . . . , b )( b + 1 , b + 2 , . . . , b + b ) . . . ( k − b a , . . . , k )we see that e i ⊗ . . . ⊗ e i k is fixed by σ if and only if it is of the form e ⊗ b i ⊗ e ⊗ b i ⊗ . . . ⊗ e ⊗ b a i a , giving d a choices for the indices i , i , . . . , i a .2) In fact this is the trace form of the image Σ k ( V ) of F [ S k ] in theoperators on V ⊗ m , dim V = d . Since Σ k ( V ) is semisimple its trace form isnon degenerate. Corollary 1.11. I ) P = X λ ⊢ k, ht ( λ ) ≤ d s λ ( d ) χ λ , II ) d c ( ρ ) = X λ ⊢ k, ht ( λ ) ≤ d s λ ( d ) χ λ ( ρ ) . (13) Proof.
This is immediate from Formula (2).5e thus have, with ht ( µ ) the number of parts of µ (cf. page 5), that P := X ρ ∈ S k d c ( ρ ) ρ = X µ ⊢ k d ht ( µ ) C µ (14)is an element of the center of the algebra Σ k ( V ) which we can thus write P = X λ ⊢ k, ht ( λ ) ≤ d s λ ( d ) χ λ = X ρ ∈ S k d c ( ρ ) ρ = X λ ⊢ k, ht ( λ ) ≤ d r λ ( d ) e λ (15)and we have: Proposition 1.12. r λ ( d ) = Y u ∈ λ ( d + c u ) . (16) Proof.
By Formula (10) and the orthogonality of characters we have: I ) e λ = χ λ (1) k ! X σ ∈ S k χ λ ( σ ) σ, II ) χ λ ( e µ ) = ( χ λ (1) if λ = µ λ = µ . (17)One has thus, from Formulas (13) I ) and (17) II) and denoting by ( χ λ , P )the usual scalar product of characters: r λ ( d ) = χ λ ( P ρ d c ( ρ ) ρ ) χ λ (1) = k !( χ λ , P ) χ λ (1) = k ! s λ ( d ) χ λ (1) (4) = Y u ∈ λ ( d + c u ) . Corollary 1.13.
The element P ρ d c ( ρ ) ρ is invertible in Σ k ( V ) with inverse ( X ρ ∈ S k d c ( ρ ) ρ ) − = X λ ⊢ k, ht ( λ ) ≤ d ( Y u ∈ λ ( d + c u )) − e λ . (18)As we shall see in § P ρ ∈ S k d c ( ρ ) ρ ) − where k is fixed and d is a parameter. We can thus use formula (18) for d ≥ k andfollowing Collins [3] we write( X ρ ∈ S k d c ( ρ ) ρ ) − = X ρ ∈ S k Wg ( d, ρ ) ρ := Wg ( d, k ) . (19)Since Wg ( d, ρ ) is a class function it depends only on the cycle partition µ = c ( ρ ) of ρ , so we may denote it by Wg ( d, µ ).From definition (12) C µ = P c ( ρ )= µ ρ we can rewrite, d ≥ kC µ = X ρ ∈ S k | c ( ρ )= µ ρ, Wg ( d, k ) = ( X ρ ∈ S k d c ( ρ ) ρ ) − = X µ ⊢ k Wg ( d, µ ) C µ . (20)6ubstituting e λ in formula (18) with its expression of Formula (17) e λ = χ λ (1) k ! X σ ∈ S k χ λ ( σ ) σ = Y u ∈ λ h − u X σ ∈ S k χ λ ( σ ) σWg ( d, k ) := X ρ ∈ S k Wg ( d, ρ ) ρ = X λ ⊢ k Y u ∈ λ h u ( d + c u ) X τ χ λ ( τ ) τ (21) Theorem 1.14. Wg ( d, σ ) = X λ ⊢ k Y u ∈ λ h u ( d + c u ) χ λ ( σ ) = X λ ⊢ k χ λ (1) χ λ ( σ ) k ! s λ ( d ) . (22) In particular Wg ( d, σ ) is a rational function of d with poles at the integers − k + 1 ≤ i ≤ k − of order p at i , p ( p + | i | ) ≤ k .Proof. We only need to prove the last estimate. By symmetry we may assumethat i ≥ p th entry of i is placed at the lower right corner of arectangle of height p and length i + p (cf. Figure at page 8). Hence if λ ⊢ k ,we have i ( p + i ) ≤ k and the claim. Formula (22), although explicit, is a sum with alternating signs so that it isnot easy to estimate a given value or even to show that it is nonzero.For σ = (1 , , . . . , k ) a full cycle a better Formula is available. FirstFormula (23) by Formanek when k = d , and then by Collins Formula (24)in general.When k = d we write Wg ( d, σ ) = a σ and then: d ! a σ = ( − d +1 d d − = 0 . (23)Collins extends Formula (23) to the case Wg ( d, σ ) getting: Wg ( d, σ ) = ( − k − C k − Y − k +1 ≤ j ≤ k − ( d − j ) − (24)with C i := (2 i )!( i +1)! i ! = i +1 (cid:0) ii (cid:1) the i th Catalan number. Which, since C d − = (2 d − d !( d − , Y − d +1 ≤ j ≤ d − ( d − j ) = (2 d − k = d , with Formanek.In order to prove Formula (24) we need the fact that χ λ ( σ ) = 0 exceptwhen λ = ( a, k − a ) is a hook partition , with the first row of some length a, ≤ a ≤ k and then the remaining k − a rows of length 1.7his is an easy consequence of the Murnaghan–Nakayama formula, see[16].In this case we have χ λ ( σ ) = ( − k − a . We thus need to make explicitthe integers s λ ( d ) , χ λ (1) for such a hook partition.For λ = ( a, k − a ), we get that the boxes are u = (1 , j ) , j = 1 , . . . , a, c u = j − , h u = ( k if j = 1 a − j + 1 if j = 1 u = ( i + 1 , , i = 1 , . . . , k − a, c u = − i, h u = k − a − i + 1 . Y u h u = k a Y j =2 ( a − j + 1) k − a Y i =1 ( k − a − i + 1) = k ( a − k − a )! . Example 1.15. a = 8 , k = 11 , (8 , ) ⊢
11 in coordinates1,11,2 1,3 1,4 1,5 1,6 1,7 1,82,13,14,1Hooks and content:11 7 6 5 4 3 2 1321 , 0 1 2 3 4 5 6 7-1-2-3Thus we finally have, substituting in Formula (22), that Wg ( σ , d ) = k X a =1 ( − k − a k ( a − k − a )! k − a Y i =1 − a ( d − i ) − (25)= k X a =1 ( − k − a Q k − i = k − a +1 ( d − i ) Q − ai = − k +1 ( d − i ) k ( a − k − a )! Y − k +1 ≤ j ≤ k − ( d − j ) − . (26)One needs to show that k X a =1 ( − a Q k − i = k − a +1 ( d − i ) Q − ai = − k +1 ( d − i ) k ( a − k − a )! = P ka =1 ( − a Q k − i = k − a +1 i ( d − i ) Q k − i = a i ( d + i ) k !( k − P k ( d ) := 1 k ! k − X b =0 ( − b +1 (cid:18) k − b (cid:19) k − Y i = k − b ( d − i ) k − Y i = b +1 ( d + i ) = ( − k − C k − . (27)8y partial fraction decomposition we have that k − a Y i =1 − a ( d − i ) − = k − a X i =1 − a b j d − j ,b = k − a Y i =1 − a, i =0 ( − i ) − = [( − k − a ( a − k − a )!] − . Therefore the partial fraction decomposition of Wg ( σ , d ), from Formula(25), is k X a =1 k [( a − k − a )!] d + X − k +1 ≤ j ≤ k − , j =0 c j d − j . On the other hand the partial fraction decomposition of the product ofFormula (26), Y − k +1 ≤ j ≤ k − ( d − j ) − = ( − k − ( k − d + X − k +1 ≤ j ≤ k − , j =0 e j d − j . It follows that the polynomial P k ( d ) of Formula (27) is a constant C with C ( − k − ( k − = k X a =1 k [( a − k − a )!] = ⇒ C = ( − k − k X a =1 ( k − k [( a − k − a )!] . So finally we need to observe that k X a =1 ( k − k [( a − k − a )!] = 1 k k − X a =0 (cid:18) k − a (cid:19) = 1 k (cid:18) k − k − (cid:19) = C k − . In fact n X a =0 (cid:18) na (cid:19) = (cid:18) nn (cid:19) as one can see simply noticing that a subset of n elements in 1 , , . . . , n distributes into a numbers ≤ n and the remaining n − a which are > n . For a partition µ ⊢ k we have defined, in Formula (12) C µ := P σ | π ( σ )= µ σ. Clearly we have for a sequence of partitions µ , µ , . . . , µ i C µ C µ . . . C µ i = X µ ⊢ k A [ µ ; µ , µ , . . . , µ i ] C µ (28)where A [ µ ; µ , µ , . . . , µ i ] counts the number of times that a product of i permutations σ , σ , . . . , σ i of types µ , µ , . . . , µ i give a permutation σ oftype µ . These numbers are classically called connection coefficients .9 emark . Notice that this number depends only on µ and not on σ .Set: A [ µ, i, h ] := X µ ,µ ,...,µ i | µ j =1 k P ij =1 ( k − ht ( µ j ))= h A [ µ ; µ , µ , . . . , µ i ] (29) A [ µ, h ] := h X i =1 ( − i A [ µ, i, h ] . Remark . For a permutation σ ∈ S k with π ( σ ) = µ we will write | σ | = | µ | := k − ht ( µ ) . (30)This is the minimum number of transpositions with product σ .We have | στ | ≤ | σ | + | τ | , see Stanley [18] p.446 for a poset interpretation.From Formula (22) we know that each Wg ( σ, d ) is a rational function of d with poles in 0 , ± , ± , . . . , ± ( k −
1) of order < k , so we can expand it ina power series in d − converging for d > k − Theorem 1.18 ([3] Theorem 2.2) . We have an expansion for ( P ρ ∈ S k d c ( ρ ) ρ ) − as power series in d − : = d − k (1 + X µ ⊢ k ∞ X h = | µ | d − h A [ µ, h ] C µ ) (31) Proof.
Recall that we denote by | µ | := k − ht ( µ ), (30). P = X ρ ∈ S k d c ( ρ ) ρ = d k (1 + X µ ⊢ k | µ =1 k d − ( k − ht ( µ )) C µ ) = d k (1 + X µ ⊢ k | µ =1 k d −| µ | C µ )so P − = d − k (1 + ∞ X i =1 ( − i ( X µ ⊢ k | µ =1 k d −| µ | C µ ) i )= d − k (1 + ∞ X i =1 ( − i ( X µ ,µ ,...,µ i | µ j =1 k d − P ij =1 | µ j | C µ C µ . . . C µ i )= d − k (1 + X µ ⊢ k ( ∞ X i =1 ( − i X µ ,µ ,...,µ i | µ j =1 k d − P ij =1 | µ j | A [ µ ; µ , µ , . . . , µ i ]) C µ )= d − k (1 + X µ ⊢ k ∞ X h = | µ | d − h A [ µ, h ] C µ )since µ + µ + . . . + µ i = µ implies | µ | ≤ P ij =1 | µ j | .10 emark . We want to see now, for µ = 1 k that the series P ∞ h = | µ | d − h A [ µ, h ]starts with h = | µ | , i.e. A [ µ, | µ | ] = 0. Thus we compute the leading coefficient A [ µ, | µ | ] which gives the asymptotic behaviour of Wg ( σ, d ).Let us denote by C [ µ ] := A [ µ, | µ | ] = ⇒ lim d →∞ d k + | σ | Wg ( σ, d ) = C [ µ ] . (32)From Formula (23) we have C [( k )] = ( − k − C k − (Catalan number) and afurther and more difficult Theorem of Collins states Theorem 1.20. [[3] Theorem 2.12 (ii)] C [( k )] = ( − k − C k − , C [( a , a , . . . , a i )] = i Y j =1 C [( a j )] . (33)Fixing σ ∈ S k with π ( σ ) = µ we have that A [ µ ; µ , µ , . . . , µ i ] is also thenumber of sequences of permutations σ j , π ( σ j ) = µ j with σ = σ σ . . . σ i .So we shall also use the notation, for π ( σ ) = µ : A [ σ ; µ , µ , . . . , µ i ] = A [ µ ; µ , µ , . . . , µ i ] , C [ σ ] := A [ σ, | σ | ] . Thus C [ µ ] = A [ µ, | µ | ] = X i =1 ( − i X µ ,µ ,...,µ i | µ j =1 k P ij =1 | µ j | = | µ | A [ µ ; µ , µ , . . . , µ i ] (34)We call a coefficient A [ µ ; µ , µ , . . . , µ i ] with µ , µ , . . . , µ i | µ j = 1 k , and P ij =1 | µ j | = | µ | a top coefficient . Q [ S k ]The study of C [ µ ] can be formulated in terms of a degeneration: Q [ ˜ S k ] ofthe multiplication in the group algebra whose elements now denote by ˜ σ .Define a new (still associative) multiplication by Q [ ˜ S k ] := ⊕ σ ∈ S k Q ˜ σ, ˜ σ ˜ σ := ( ] σ σ if | σ σ | = | σ | + | σ | . (35)Contrary to the semisimple algebra Q [ S k ] the algebra Q [ ˜ S k ] is a gradedalgebra, with Q [ ˜ S k ] h = ⊕ σ ∈ S k | | σ | = h Q ˜ σ and has I := ⊕ σ ∈ S k | σ =1 Q ˜ σ = ⊕ k − h =1 Q [ ˜ S k ] h I have made a considerable effort trying to understand, and hence verify, the proof ofthis Theorem in [3], to no avail. I hope somebody has verified it. At any rate the Theoremis true as I will show presently with a simple natural proof.
11s a nilpotent ideal, I k = 0, its nilpotent radical. Observe that | σ σ | = | σ | + | σ | ⇐⇒ c ( σ σ ) = c ( σ ) + c ( σ ) − k so if c ( σ ) + c ( σ ) ≤ k we know a priori that the product ˜ σ ˜ σ = 0.In this algebra the multiplication of two elements ˜ C µ , ˜ C µ associated toconjugacy classes as in (12) involves only the top coefficients and is:˜ C µ ˜ C µ = X | µ | = | µ | + | µ | A [ µ ; µ , µ ] ˜ C µ . (36)We then have( X ρ ∈ S k d c ( ρ ) ˜ ρ ) − = d − k (1 + X µ ⊢ k | µ =1 k d −| µ | ˜ C µ ) − = d − k (1 + X µ ⊢ k d −| µ | C [ µ ] ˜ C µ )= d − k (1 + k − X h =1 d − h ( X µ ⊢ k || µ | = h C [ µ ] ˜ C µ )) . (37)Notice that if h = k − µ with | µ | = k − µ = ( k ) thepartition of the full cycle.Hence in Formula (37) the lowest term is d − k +1 C [( k )] ˜ C ( k ) .An example,which the reader can skip, the connection coefficients for S ,in box the top ones (write the elements C µ with lowercase): c , , c , c , c c , , c , , , + 3 c , + 2 c , c , , + 4 c c , , + 2 c c , + 4 c , c , c , , + 4 c c , , , + 4 c , + 8 c , c , c , , + 4 c c , c , , + 2 c c , c , , , + 2 c , c , , + c c c , + 4 c , c , , + 4 c c , , + c c , , , + 3 c , + 2 c , Setting a = c , , , b = c , , c = c , , d = c compute (37) a = 3 b + 2 c, ab = 4 d, ac = 2 dP = 1 + T, T = x − a + x − ( b + c ) + x − d, (1 + T ) − = 1 − T + T − T T = x − a +2 x − a ( b + c ) = x − (3 b +2 c )+ x − d, T = x − a (3 b +2 c ) = x − (12+4) d = x − d − T + T − T = − x − a − x − ( b + c ) − x − d + x − (3 b + 2 c ) + x − d − x − d = − x − a + x − (2 b + c ) − x − d The conjugacy classes and their cardinality in S : (cid:0) , c , , , , , c , , , , c , , , c , , , c , , c , , c (cid:1) S . The numbers to theright are the degrees | µ | : a = c , , , , b = c , , , c = c , , , d = c , , e = c , , f = c , c , , , c , , c , , c , c , c c , , , c , , + 2 c , , c , + c . c , + 3 c . c c c , , c , + c . c c c , , c , + 3 c . c c c , c c , c c a = 3 b +2 c, ab = 4 d + e, ac = 2 d +3 e, ad = 5 f, ae = 5 f, b = 5 f, bc = 5 f, c = 5 f, T, T = x − a + x − ( b + c ) + x − ( d + e ) + x − fT = x − a + x − ( b + c ) + 2 x − a ( b + c ) + 2 x − a ( d + e )= x − (3 b + 2 c ) + 2 x − (6 d + 4 e ) + 40 x − fT = x − a (3 b + 2 c ) + 2 x − a (6 d + 4 e ) + x − ( b + c )(3 b + 2 c )= x − (12 d + 3 e + 4 d + 6 e ) + x − (100 + 15 + 10 + 15 + 10) f = x − (16 d + 9 e ) + x − fT = x − a (16 d + 9 e ) = x − (16 · f = x − f −
150 + 40 − C i =Catalan(i): 1, 2, 5, 14, 42,. . . Catalan(4)=14! − T + T − T + T = − ( x − a + x − ( b + c )+ x − ( d + e ))+ x − (3 b +2 c )+2 x − (6 d +4 e ) − x − (16 d +9 e )+14 f = − x − a − x − ( b + c ) − x − ( d + e )+ x − (3 b +2 c )+2 x − (6 d +4 e ) − x − (16 d +9 e )= − x − a + x − (3 b + 2 c − b − c ) + x − (12 d + 8 e − d − e − d − e )= − x − a + x − (2 b + c ) + x − ( − d − e ) + 14 f. .20.2 Young subgroups Let Π := { A , A , . . . , A j } , | A i | = a i be a decomposition of the set [1 , , . . . , k ]:i.e. A ∪ A ∪ . . . ∪ A j = [1 , , . . . , k ] , A i ∩ A j = ∅ , ∀ i = j. Definition .
1. The subgroup of S k fixing this decomposition is theproduct Q ji =1 S A i = Q ji =1 S a i of the symmetric groups S a i . It is usuallycalled a Young subgroup and will be denoted by Y Π .2. Given two decompositions of [1 , , . . . , k ], Π := { A , A , . . . , A j } , andΠ := { B , B , . . . , B h } we say that Π ≤ Π if each set A i is containedin one of the sets B d . This is equivalent to the condition Y Π ⊂ Y Π .3. In particular, if σ ∈ S k we denote by Π σ the decomposition of [1 , , . . . , k ]induced by its cycles and denote Y σ := Y Π σ . Remark . Observe that τ ∈ Y Π if and only if Π τ ≤ Π. The conjugacyclasses of Y Π are the products of the conjugacy classes in the blocks A i .Then we have for the group algebra and τ = ( τ , τ , . . . , τ j ) ∈ Y Π : Q [ Y Π ] = ⊗ ji =1 Q [ S a i ] ⊂ Q [ S k ] , ( τ , τ , . . . , τ j ) = τ ⊗ τ ⊗ . . . ⊗ τ j . (38)We denote by c τ the sum of the elements of the conjugacy class of τ in Y Π in order to distinguish it from C τ the sum over the conjugacy class in S k .We have: τ = ( τ , τ , . . . , τ j ) ∈ Y Π , c τ (12) = C τ ⊗ C τ ⊗ . . . ⊗ C τ j . (39)The first remark is: Remark . If τ = ( τ , τ , . . . , τ j ) ∈ Y Π then c ( τ ) = c ( τ ) + c ( τ ) + · · · + c ( τ j ) , = ⇒ | τ | = X i a i − c ( τ ) = X i ( a i − c ( τ i )) = | τ | + | τ | + · · · + | τ j | . (40)As a consequence if γ = ( γ , γ , . . . , γ j ) , τ = ( τ , τ , . . . , τ j ) ∈ Y Π we have | γτ | = | γ | + | τ | ⇐⇒ | γ i τ i | = | γ i | + | τ i | , ∀ i. (41)If we then consider the associated discrete algebras, From Formulas (41)and (38) we deduce an analogous of Formula (38) for the discrete algebras: Q [ ˜ Y Π ] = ⊗ ji =1 Q [ ˜ S a i ] ⊂ Q [ ˜ S k ] , τ = ( τ , τ , . . . , τ j ) , ˜ τ = ˜ τ ⊗ ˜ τ ⊗ · · · ⊗ ˜ τ j . (42)Formula (40) tells us that Q [ ˜ Y Π ] = ⊗ ji =1 Q [ ˜ S a i ] as graded tensor product andthe inclusion in Q [ ˜ S k ] preserves the degrees.14 .23.1 A proof of Theorem 1.20 In particular let σ ∈ S k and σ = c c . . . c j its cycle decomposition.Let A i be the support of the cycle c i of σ and a i its cardinality, so thatΠ σ = { A , . . . , A j } and Y σ = Y Π σ . We have σ ∈ Y σ and its conjugacy classin Y σ is the product of the conjugacy classes of the cycles ( a i ) ⊂ S a i , (12).We denote, as before, by c σ the sum of the elements of this conjugacy class.We have now a very simple but crucial fact; Proposition 1.24.
1. Let ( i, i , . . . , i a ) , ( j, j , . . . , j b ) be two disjoint cy-cles and take the transposition ( i, j ) then: ( i, i , . . . , i a )( j, j , . . . , j b )( i, j ) = ( i, j , . . . , j b , i , . . . , i a , j ) (43)( i, j )( i, i , . . . , i a )( j, j , . . . , j b ) = ( j, j , . . . , j b , i , . . . , i a , i ) (44)
2. Let σ ∈ S k and τ = ( i, j ) a transposition. Then | στ | = | τ σ | = | σ | ± and | στ | = | τ σ | = | σ | − if and only if the two indices i, j both belongto one of the sets of the partition of σ , i.e. τ = ( i, j ) ∈ Y σ .Proof.
1) is clear and 2) follows immediately from 1).Notice that Π στ < Π σ and is obtained from Π σ by replacing the supportof the cycle in which i, j appear with two subsets support of the 2 cycles inwhich this splits. Similarly for Π τσ .From this we deduce the essential result of this section: Corollary 1.25.
Let σ ∈ S k . Consider a decomposition σ = σ σ . . . , σ h ,σ i ∈ S k , σ i = 1 , ∀ i with | σ | = | σ | + | σ | + . . . + | σ h | . Then for all i we have σ i ∈ Y Π σ = Y σ (Definition 1.21).Proof. By induction on h , if h = 1 there is nothing to prove.If σ = ( i, j ) is a transposition then the the claim follows by inductionon τ σ = ¯ σ = σ . . . , σ h , and Proposition 1.24.If | σ | > σ = τ ¯ σ with | ¯ σ | = | σ | − σ ∈ S k and σ = c c . . . c j its cycle decomposition. Let A i be thesupport of the cycle c i and a i its cardinality, so that Π σ = { A , . . . , A j } .By the previous Corollary 1.25 and Remark 1.16 the contribution to σ in the terms of Formula (29) are all in the subgroup Y σ so that finally C [ σ ] = C [˜ σ ] with C [˜ σ ] computed in Q [ ˜ Y σ ] . In order to compute C [˜ σ ] we observe that the term d − k −| ˜ σ | C [˜ σ ] c ˜ σ = d − k −| σ | C [˜ σ ] c ˜ σ is the lowest term in d − in( X ρ ∈ Y σ d c ( ρ ) ˜ ρ ) − = j O i =1 ( X ρ ∈ S ai d c ( ρ ) ˜ ρ ) − . (45)15rom Formula (37) applied to the various full cycles c i ∈ S a i we have thatthe lowest term in ( P ρ ∈ S ai d c ( ρ ) ˜ ρ ) − is d − a i +1 C [( a i )] C ( a i ) so that we havefinally that the lowest term in Formula (45) is d − k −| σ | C [˜ σ ] c ˜ σ (39) = j Y i =1 d − a i +1 C [( a i )] C ( a ) ⊗ . . . ⊗ C ( a j ) , = ⇒ C [ σ ] = C [˜ σ ] = j Y i =1 C [( a i )] (23) = j Y i =1 ( − a i − C a i − . (46)We have proved, Formula (23) that ( − a i − C [( a i )] is the Catalan number C a i − and the proof of Theorem 1.20 is complete. The case k = d is of special interest, see § Wg ( d, µ ) = a µ sothat P µ ⊢ d Wg ( d, µ ) c µ = P µ a µ c µ in Formula (20).A computation using Mathematica gives d ≤ d ! P µ ⊢ d a µ c µ : − c + 43 c , − c , + 35 c + 2110 c − c , − c + 2235 c + 2935 c , + 13435 c . − c , − c , − c , + 59 c + 115126 c , + 8063 c , + 14518 c − c , − c , − c , , − c − c + 300539 c , + 9221617 c , + 2633 c , + 23961617 c , + 1180539 c , + 10508539 c − c , − c , − c , , − c , − c , − c , − c , + 713 c + 10151716 c , + 13791716 c , + 259312 c , , + 21071716 c , + 94013432 c , + 73851716 c , + 1848493432 c c − c , + 144341485 c , + 11282819305 c , , − c , − c , , + 4133219305 c , − c , + 86086435 c , + 2471819305 c , − c , + 1712219305 c , , − c , , − c , ,
16 151195 c , + 124195 c − c , − c , + 11862145 c , + 7991485 c , + 7961485 c − c the reader will notice certain peculiar properties of these sequences. Conjecture
There is a sign change at the cycle, in the sense that for n odd W g (( n )) is the smallest of the positive values and for n even W g (( n )) isthe biggest of the negative values. The negatives are strictly increasing in thelexicographic order the positives are strictly decreasing in the lexicographicorder.I verified this up to d = 14.These deserve further investigation, maybe the factorization of Jucys: X ρ ∈ S k d c ( ρ ) ρ = k Y i =1 ( d + J i ) , J i = (1 , i ) + (2 , i ) + . . . + ( i − , i )see [10] and [14] can be used. ( V M ∗ d ) G Preliminary to the next step we need to recall the theory of antisymmetricconjugation invariant functions on M d . This is a classical theory over a fieldof characteristic 0 which one may take as Q .First, let U be a vector space, a polynomial g ( x , . . . , x m ) in m variables x i ∈ U is antisymmetric or alternating in the variables X := { x , . . . , x m } iffor all permutations σ ∈ S m we have g ( x σ (1) , . . . , x σ ( m ) ) = ǫ σ g ( x , . . . , x m ) , ǫ σ the sign of σ. A simple way of forming an antisymmetric polynomial from a given one g ( x , . . . , x m ) is the process of alternation Alt X g ( x , . . . , x m ) := X σ ∈ S m ǫ σ g ( x σ (1) , . . . , x σ ( m ) ) . (47)Recall that the exterior algebra V U ∗ , with U a vector space, can be thoughtof as the space of multilinear alternating functions on U . Then exteriormultiplication as functions is given by the Formula: f ( x , . . . , x h ) ∈ h ^ U ∗ ; g ( x , . . . , x k ) ∈ k ^ U ∗ ,f ∧ g ( x , . . . , x h + k ) = 1 h ! k ! X σ ∈ S h + k f ( x σ (1) , . . . , x σ ( h ) ) g ( x σ ( h +1) , . . . , x σ ( h + k ) )(48)= 1 h ! k ! Alt x ,...,x h + k f ( x , . . . , x h ) g ( x h +1 , . . . , x h + k ) ∈ h + k ^ U ∗ . (49)It is well known that: we avoid on purpose multiplying by 1 /m ! roposition 1.27. A multilinear and antisymmetric polynomial g ( x , . . . , x m ) in m variables x i ∈ C m is a multiple, a det( x , . . . , x d ) , of the determinant.In fact if the polynomial has integer coefficients a ∈ Z . For a multilinear and antisymmetric polynomial map g ( x , . . . , x m ) ∈ U to a vector space, each coordinate has the same property so g ( x , . . . , x m ) = det( x , . . . , x m ) a, a ∈ U. We apply this to U = M d . Let us identify M d = C d using the canonicalbasis of elementary matrices e i,j ordered lexicographically e.g.: d = 2 , e , , e , , e , , e , . Given d matrices Y , . . . , Y d ∈ M d we may consider them as elementsof C d and then form the determinant det( Y , . . . , Y d ).By Proposition 1.27 the 1 dimensional space V d M ∗ d has as generatorthe determinant det( Y , . . . , Y d ) which, since the conjugation action by G := GL ( d, Q ) on M d is by transformations of determinant 1, is thus an invariantunder the action by G .The theory of G invariant antisymmetric multilinear G invariant func-tions on M d is well known and related to the cohomology of G .The antisymmetric multilinear G invariant functions on M d form thealgebra ( V M ∗ d ) G . This is a subalgebra of the exterior algebra V M ∗ d and canbe identified to the cohomology of the unitary group. As all such cohomologyalgebras it is a Hopf algebra and by Hopf’s Theorem it is the exterior algebragenerated by the primitive elements.The primitive elements of ( V M ∗ d ) G are, see [11]: T i − = T i − ( Y , . . . , Y i − ) := tr ( St i − ( Y , . . . , Y i − )) (50) St i − ( Y , . . . , Y i − ) = X σ ∈ S i − ǫ σ Y σ (1) . . . Y σ (2 i − with i = 1 , . . . , d . In particular, since these elements generate an exterioralgebra we have: Remark . A product of elements T i is non zero if and only if the T i involved are all distinct, and then it depends on the order only up to a sign.The 2 n different products form a basis of ( V M ∗ d ) G . The non zero productof all these elements T i − ( Y , . . . , Y i − ) is in dimension d . We denote T d ( Y , Y , . . . , Y d ) = T ∧ T ∧ T ∧ · · · ∧ T d − . (51) Proposition 1.29.
A multilinear antisymmetric function of Y , . . . , Y d isa multiple of T ∧ T ∧ T ∧ · · · ∧ T d − .Remark . The function det( Y , . . . , Y d ) is an invariant of matrices soit must have an expression as in Formula (6). In fact up to a computableinteger constant [6] this equals the exterior product of Formula (51)18he constant of the change of basis when we take as basis the matrixunits can be computed up to a sign, see [6]: T d ( Y ) = C d det( Y , . . . , Y d ) , C d := ± · · · (2 d − · · · ( d − . (52) Rather than following the historical route we shall first discuss the paper ofCollins, since this will allow us to introduce some notations useful for thediscussion of Formanek’s results.
In the paper [3], Collins introduces the Weingarten function in the followingcontext. He is interested in computing integrals of the form Z U ( d ) k Y ℓ =1 u j ℓ ,h ℓ k Y m =1 ¯ u i m ,p m du where U ( d ) is the unitary group of d × d matrices and the elements u i,j the entries of a matrix X ∈ U ( d ) while ¯ u j,i the entries of X − = U ∗ = ¯ U t .Here du is the normalized Haar measure. If one translates by a scalar matrix α, | α | = 1 then the integrand is multiplied by α k ¯ α k , on the other handHaar measure is invariant under multiplication so that this integral vanishesunless we have k = k . In this case the computation will be algebraic basedon the following considerations.Let us first make some general remarks. A finite dimensional represen-tation R of a compact group G (with the dual denoted by R ∗ ), decomposesinto the direct sum of irreducible representations. In particular if R G de-notes the subspace of G invariant vectors there is a canonical G equivariantprojection E : R → R G . The projection E can be written as integral E ( v ) := Z G g · v dg, dg normalized Haar measure. (53)In turn the integral E ( v ) = R G g · v dg is defined in dual coordinates by h ϕ | E ( v ) i = h ϕ | Z G g · v dg i := Z G h ϕ | g · v i dg, ∀ ϕ ∈ R ∗ . (54)The functions, of g ∈ G , h ϕ | g · v i , ϕ ∈ R ∗ , v ∈ R are called representativefunctions ; therefore an explicit formula for E is equivalent to the knowl-edge of integration of representative functions. In fact usually the integralis computed by some algebraic method of computation of E .In the case of V = C d with natural basis e i and dual basis e j .19e take R = End ( V ) with the conjugation action of GL ( V ) or of itscompact subgroup U ( d ) of unitary d × d matrices: Xe h,p X − = X i,j u i,h ¯ u j,p e i,j , X = X i,j u i,j e i,j ∈ U ( d ) , X − = X i,j ¯ u j,i e i,j . A basis of representative functions for R = End ( V ) is tr ( e i,j Xe h,p X − ) = tr ( e i,j X a,b u a,h ¯ u b,p e a,b ) = u j,h ¯ u i,p , i, j, h, p = 1 , . . . , d. (55)Since a duality between End ( V ) ⊗ k and itself is the non degenerate pairing: h A | B i := tr ( A · B )a basis of representative functions of End ( V ) ⊗ k is formed by the products tr ( e i ,j ⊗ e i ,j . . . ⊗ e i k ,j k · Xe h ,p X − ⊗ Xe h ,p X − . . . ⊗ Xe h k ,p k X − ) = tr (cid:16) e i,j · X e h,p X − (cid:17) = k Y ℓ =1 tr ( e i ℓ ,j ℓ · Xe h ℓ ,p ℓ X − ) = k Y ℓ =1 u j ℓ ,h ℓ ¯ u i ℓ ,p ℓ , (56)where in order to have compact notations we write i := ( i , i , . . . , i k ) , e i,j = e i ,j ⊗ e i ,j . . . ⊗ e i k ,j k (57)Collins writes the explicit Formula (64) for Z U ( d ) k Y ℓ =1 u j ℓ ,h ℓ ¯ u i ℓ ,p ℓ du = Z U ( d ) tr (cid:16) e i,j · X e h,p X − (cid:17) dX = tr (cid:16) e i,j · E ( e h,p ) (cid:17) (58)In order to do this, it is enough to have an explicit formula for the equivariantprojection E of End ( V ) ⊗ k to the GL ( V ) (or U ( d )) invariants Σ k ( V ), thealgebra generated by the permutation operators σ ∈ S k acting on V ⊗ k .His idea is to consider first the mapΦ : End ( V ) ⊗ k → Σ k ( V ) , Φ( A ) := X σ tr ( A ◦ σ − ) σ. (59)This map is a GL ( V ) equivariant map to Σ k ( V ) , but it is not a projection.In fact restricted to Σ k ( V ) , we haveΦ : Σ k ( V ) → Σ k ( V ) , Φ( τ ) := X σ ∈ S k tr ( τ ◦ σ − ) σ. Setting σ = γτ, τ σ − = γ − we have:Φ( τ ) = X γ ∈ S k tr ( γ − ) γ τ = Φ(1) τ = τ Φ(1) = τ X γ ∈ S k tr ( γ − ) γ. (60)20e have seen, in Corollary 1.13, thatΦ(1) = X γ ∈ S k tr ( γ − ) γ = X γ ∈ S k d c ( γ ) γ is a central invertible element of Σ k ( V ). So the equivariant projection E is Φ composed with multiplication by the inverse Wg ( d, k ) of the elementΦ(1) = P γ ∈ S k tr ( γ − ) γ given by Formula (22) or (18). E = ( X γ ∈ S k tr ( γ − ) γ ) − ◦ Φ = Φ(1) − ◦ Φ = Wg ( d, k ) ◦ Φ . (61)Of course Φ( e h,p ) = X σ tr ( e h,p ◦ σ − ) σ = ⇒ E ( e h,p ) = X γ ∈ S k Wg ( d, γ ) γ X σ tr ( e h,p ◦ σ − ) σ and Formula (58) becomes tr ( e i,j ◦ X γ ∈ S k Wg ( d, γ ) γ X σ tr ( e h,p ◦ σ − ) σ ) (62)= X γ,σ ∈ S k tr ( e i,j ◦ γ ) tr ( e h,p ◦ σ − ) Wg ( d, γσ − ) (63)From Formulas (7) and (8) since e i,j = e i ⊗ e j we have tr ( e i ,j ⊗ e i ,j . . . ⊗ e i k ,j k ◦ γ ) = Y h h e i γ ( h ) | e j h i = Y h δ j h i γ ( h ) (63) = X γ,σ ∈ S k Y ℓ δ j ℓ i γ ( ℓ ) Y ℓ δ p σ ( ℓ ) h ℓ Wg ( d, γσ − ) = X γ,σ ∈ S k δ jγ ( i ) δ σ ( p ) h Wg ( d, γσ − ) . (64) Remark . In particular for i ℓ = h ℓ = p ℓ = ℓ and j ℓ = τ ( ℓ ) , ≤ ℓ ≤ k ,Formula (64) gives Wg ( d, τ ).Collins then goes several steps ahead since he is interested in the asymp-totic behaviour of this function as d → ∞ and proves an asymptotic expres-sion for any σ in term of its cycle decomposition, Theorem 1.20. In work in progress with Felix Huber, [9], we consider the problem of under-standing tensor valued polynomials of k , d × d matrices.That is maps from n tuples of d × d matrices x , . . . , x n ∈ End ( V ) totensor space End ( V ) ⊗ k of the form G ( x , . . . , x n ) = X i α i m ,i ⊗ m ,i ⊗ . . . ⊗ m k,i , α i ∈ C m j,i monomials in the x i . A particularly interesting case is when the polynomial is multilinear andalternating in n = d matrix variables.In this case, by Proposition 1.27 we have21 heorem 2.4. G ( x , . . . , x d ) = det( x , . . . , x d ) ¯ J G .
2. Moreover we have the explicit formula G ( e , , e , , e , , e , , . . . , e d,d ) = ¯ J G .
3. The element ¯ J G ∈ M ⊗ kd is GL ( k ) invariant and so ¯ J G ∈ Σ k ( V ) is alinear combinations of the elements of the symmetric group S n ⊂ M ⊗ kd given by the permutations. For theoretical reasons instead of computing ¯ J G it is better to computeits multiple, as in Formula (52): G ( x , . . . , x d ) = T d ( X ) J G , ¯ J G = C d J G . (65)Using Formula (59) we may first computeΦ( G ( x , . . . , x d )) = X σ ∈ S k tr ( σ − ◦ G ( x , . . . , x d )) = T d ( X )Φ( J G ) . Consider the special case G d ( Y , . . . , Y d ) := Alt Y ( m ( Y ) ⊗ · · · ⊗ m d ( Y )) , m i ( Y ) = Y ( i − +1 . . . Y i . (66) Lemma 2.5.
Alt Y tr ( σ − ◦ m ( Y ) ⊗ · · · ⊗ m d ( Y )) = ( T d ( Y ) if σ = 10 otherwise (67) Proof. tr ( σ − ◦ m ( Y ) ⊗ · · · ⊗ m d ( Y )) = j Y i =1 tr ( N i )with N i the product of the monomials m j for j in the i th cycle of σ , cf.Formula (6). The previous invariant gives by alternation the invariant Alt
Y j Y i =1 tr ( N i ) = T a ∧ T a ∧ · · · ∧ T a j , a i = degree of N i in degree d . If σ = 1 we have j < d hence the product is 0, since the onlyinvariant alternating in this degree is T ∧ T ∧ T ∧ . . . ∧ T d − .On the other hand if σ = 1 we have N i = m i and the claim follows. Proposition 2.6.
We have G d ( Y , . . . , Y d ) := Alt Y ( m ( Y ) ⊗ · · · ⊗ m d ( Y )) = T d ( Y ) Wg ( d, d ) . (68) Proof.
The previous Lemma in fact implies that Φ( G d ( Y , . . . , Y d )) = T d ( Y )1 d therefore Φ( J G d ) (60) = Φ(1) J G d = 1 so that J G d = Φ(1) − = Wg ( d, d ).22 .7 The construction of Formanek Let us now discuss a theorem of Formanek relative to a conjecture of Regev,see [6] or [1], that a certain explicit central polynomial F ( X, Y ) in d , d × d matrix variables X = { X , . . . , X d } and another d , d × d matrix variables Y = { Y , . . . , Y d } is non zero. This polynomial plays an important role inthe theory of polynomial identities, see [1].The definition of F ( X, Y ) is this, decompose d = 1+3+5+ . . . +(2 d − d variables X and the d variables Y in thetwo lists. Construct the monomials m i ( X ) , i = 1 , . . . , d and similarly m i ( Y )as product in the given order of the given 2 i − X i of the i th listas for instance m ( X ) = X , m ( X ) = X X X , m ( X ) = X X X X X , . . . .m i ( X ) = X ( i − +1 . . . X i , m i ( Y ) = Y ( i − +1 . . . Y i . We finally define F ( X, Y ) :=
Alt X Alt Y ( m ( X ) m ( Y ) m ( X ) m ( Y ) . . . m d ( X ) m d ( Y )) , (69)where Alt X (resp. Alt Y ) is the operator of alternation, Formula (47), in thevariables X (resp. Y ). By Theorem 2.4 it takes scalar values, a multiple of T d ( X ) T d ( Y ), but it could be identically 0. Theorem 2.8. F ( X, Y ) = ( − d − d !) (2 d − T d ( X ) T d ( Y ) Id d (70) (52) = ( − d − C d ( d !) (2 d −
1) ∆( X )∆( Y ) Id d ; ∆( X ) = det( X , . . . , X d ) . Notice that by Formula (52) the coefficient is an integer (as predicted).Thus F ( X, Y ) is a central polynomial. In fact it has also the property ofbeing in the conductor of the ring of polynomials in generic matrices insidethe trace ring. In other words by multiplying F ( X, Y ) by any invariantwe still can write this as a non commutative polynomial. This follows bypolarizing in z the identity, cf. [1] Proposition 10.4.9 page 286.det( z ) d F ( X, Y ) = F ( zX, Y ) = F ( X, zY ) = F ( Xz, Y ) = F ( X, Y z ) . Let us follow Formanek’s proof. First, since F ( x, y ) is a central polyno-mial Formula (70) is equivalent to: tr ( F ( X, Y )) = ( − d − d ( d !) (2 d − T d ( X ) T d ( Y ) . (71)23ow we have, with σ = (1 , . . . , d ) the cycle: tr ( F ( X, Y )) = tr ( σ − ◦ Alt X Alt Y ( m ( X ) m ( Y ) ⊗ m ( X ) m ( Y ) ⊗ . . . ⊗ m d ( X ) m d ( Y )) , (72) (68) = tr ( σ − ◦ Alt X ( m ( X ) ⊗ m ( X ) ⊗ . . . ⊗ m d ( X ) · Wg ( d, d )) T d ( Y ) . Denote Wg ( d, d ) = P τ ∈ S d a τ τ , we have tr ( σ − ◦ Alt X ( m ( X ) ⊗ m ( X ) ⊗ . . . ⊗ m d ( X ) · Wg ( d, d ))= X τ a τ tr ( σ − τ ◦ Alt X ( m ( X ) ⊗ m ( X ) ⊗ . . . ⊗ m d ( X ))which, by Lemma 2.5 equals a σ T d ( X ) . Therefore the main Formula (70)follows from Formula (23). If k > d of course there is still an expression as in Formula (19) but it is notunique.It can be made unique by a choice of a basis of Σ k ( V ). This may be doneas follows. Definition . Let 0 < d be an integer and let σ ∈ S n .Then σ is called d –bad if σ has a descending subsequence of length d ,namely, if there exists a sequence 1 ≤ i < i < · · · < i d ≤ n such that σ ( i ) > σ ( i ) > · · · > σ ( i d ). Otherwise σ is called d –good . Remark . σ is d –good if any descending sub–sequence of σ is of length ≤ d −
1. If σ is d -good then σ is d ′ -good for any d ′ ≥ d .Every permutation is 1-bad. Theorem 3.3. If dim( V ) = d the d + 1 –good permutations form a basis of Σ k ( V ) .Proof. Let us first prove that the d + 1–good permutations span Σ k,d .So let σ be d + 1–bad so that there exist 1 ≤ i < i < · · · < i d +1 ≤ n such that σ ( i ) > σ ( i ) > · · · > σ ( i d + 1). If A is the antisymmetrizeron the d + 1 elements σ ( i ) , σ ( i ) , · · · , σ ( i d + 1) we have that Aσ = 0 inΣ k ( V ), that is, in Σ k ( V ), σ is a linear combination of permutations ob-tained from the permutation σ with some proper rearrangement of the in-dices σ ( i ) , σ ( i ) , · · · , σ ( i d + 1).These permutations are all lexicographically < σ . One applies the same algorithm to any of these permutations whichis still d + 1–bad.This gives an explicit algorithm which stops when σ isexpressed as a linear combination of d + 1–good permutations (with integercoefficients so that the algorithm works in all characteristics).In order to prove that the d + 1–good permutations form a basis, it isenough to show that their number equals the dimension of Σ k,d . This isinsured by a classical result of Schensted which we now recall.24 .3.1 The RSK and d -good permutations The RSK correspondence , see [12], [19], is a combinatorially defined bi-jection σ ←→ ( P λ , Q λ ) between permutations σ ∈ S n and pairs P λ , Q λ ofstandard tableaux of same shape λ, where λ ⊢ n .In fact more generally it associates to a word, in the free monoid, apair of tableaux, one standard and the other semistandard filled with theletters of the word. This correspondence may be viewed as a combinatorialcounterpart to the Schur–Weyl and Young theory.The correspondence is based on a simple game of inserting a letter.We have some letters piled up so that lower letters appear below higherletters and we want to insert a new letter x . If x fits on top of the pile weplace it there otherwise we go down the pile, until we find a first place wherewe can replace the existing letter with x . We do this and expel that letter,first creating a new pile or, if we have a second pile of letters then we try toplace that letter there and so on.So let us pile inductively the word strange . e e, g ge, n nge, a nga e, r rnga e, t trnga e, s srng ta e. Notice that, as we proceed, we can keep track of where we have placed thenew letter, we do this by filling a corresponding tableau.6532 71 4 , srng ta e.
It is not hard to see that from the two tableaux one can decrypt the wordwe started from giving the bijective correspondence.Assume now that σ ←→ ( P λ , Q λ ), where P λ , Q λ are standard tableaux,given by the RSK correspondence. By a classical theorem of Schensted [17], ht ( λ ) equals the length of a longest decreasing subsequence in the permuta-tion σ . Hence σ is d + 1-good if and only if ht ( λ ) ≤ d .Now M λ has a basis indexed by standard tableaux of shape λ , see [16].Thus the algebra Σ k ( V ) has a basis indexed by pairs of tableaux of shape λ. ht ( λ ) ≤ d and the claim follows.Therefore one may define the Weingarten function for all k as a functionon the d + 1–good permutations in S k . Robinson, Schensted, Knuth .3.2 Cayley’s Ω process It may be interesting to compare this method of computing the integralsof Formula (61) with a very classical approach used by the 19 th centuryinvariant theorists.Let me recall this for the modern readers. Recall first that, given a d × d matrix X = ( x i,j ), its adjugate is V d − ( X ) = ( y i,j ) with y i,j the cofac-tor of x j,i that is ( − i + j times the determinant of the minor of X ob-tained by removIng the j row and i column. Then the inverse of X equalsdet( X ) − V d − ( X ).It is then easy to see that, substituting to u i,j the variables x i,j and to¯ u i,j the polynomial y i,j one transforms a monomial M = Q kℓ =1 u j ℓ ,h ℓ ¯ u i ℓ ,p ℓ into a polynomial π d ( M ) in the variables x i,j homogeneous of degree dk , theinvariants under U d become powers det( X ) k . Denote by S kd ( x i,j ) the spaceof these polynomials which, under the action of GL ( d ) × GL ( d ), decomposesby Cauchy formula, cf. Formula 6.18, page 178, of [1]. Then we have also anequivariant projection from these polynomials to the 1–dimensional spacespanned by det( X ) k , it is given through the Cayley Ω process used by Hilbertin his famous work on invariant theory. The Ω process is the differentialoperator given by the determinant of the matrix of derivatives: X = ( x i,j ) , Y = ( ∂∂x i,j ) , Ω := det( Y ) . (73)We have that Ω k is equivariant under the action by SL ( n ) so it maps to0 all the irreducible representations different from the 1–dimensional spacespanned by det( X ) k whileΩ det( X ) k = k ( k + 1) . . . ( k + d −
1) det( X ) k − . Both statements follow from the Capelli identity, see [16] § X )Ω = det( a i,j ) , a i,i = ∆ i,i + n − i, a i,j = ∆ i,j , i = j the polarizations ∆ i,j = d X h =1 x i,h ∂∂x h,j . If we denote by x i := ( x i, , . . . , x i,n ) we have the Taylor series for a function f ( x , . . . , x n ) of the vector coordinates x i . f ( x , . . . , x j + λx i , . . . , x n ) = ∞ X k =0 ( λ ∆ i,j ) k k ! f ( x , . . . , x n ) . Thus Z U M du = Ω k π d ( M ) Q ki =1 ( i ( i + 1) . . . ( i + d − . (74)26e can use Remark 2.2 to give a possibly useful formula: Wg ( d, γ ) = Ω k π d ( M ) Q ki =1 ( i ( i + 1) . . . ( i + d − , M = k Y i =1 u i,i ¯ u i,γ ( i ) . (75)Let me discuss a bit some calculus with these operators. Lemma 3.4. If i = j then ∆ ij commutes with Ω and with det( X ) while [∆ ii , det( X )] = det( X ) , [∆ ii , Ω] = − Ω . (76) Proof.
The operator ∆ ij commutes with all of the columns of Ω exceptthe i th column ω i with entries ∂∂x it . Now [∆ ij , ∂∂x it ] = − ∂∂x jt , from which[∆ ij , ω i ] = − ω j . The result follows immediately.Let us introduce a more general determinant, analogous to a character-istic polynomial. We denote it by C m ( ρ ) = C ( ρ ) and define it as: ∆ , + m − ρ ∆ , . . . ∆ ,m ∆ , ∆ , + m − ρ . . . ∆ ,m . . . . . . . . . . . .. . . . . . . . . . . . ∆ m − , ∆ m − , . . . ∆ m − ,m ∆ m, ∆ m, . . . ∆ m,m + ρ . We have now a generalization of the Capelli identity:
Proposition 3.5. Ω C ( k ) = C ( k + 1)Ω , det( X ) C ( k ) = C ( k −
1) det( X )det( X ) k Ω k = C ( − ( k − C ( − ( k − . . . C ( − C, Ω k det( X ) k = C ( k ) C ( k − . . . C (1) . Proof.
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