aa r X i v : . [ m a t h . C O ] F e b Magic squares and the symmetric group
Ofir Gorodetsky
Abstract
Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use thecharacteristic map to give a new combinatorial proof of their result. We also extend their computationto moments of traces of symmetric powers, where the same result holds but in a wider range.
Consider the complex unitary group U ( N ) endowed with the probability Haar measure. The n th secularcoefficient of U ∈ U ( N ) is defined through the expansiondet( zI + U ) = N X n =0 z N − n Sc n ( U ) . If A = ( a i,j ) is an m × n matrix with non-negative integer entries, Diaconis and Gamburd [DG06] definethe row-sum vector row( A ) ∈ Z m and column-sum vector col( A ) ∈ Z n byrow( A ) i = n X j =1 a i,j , col( A ) j = m X i =1 a i,j . Given two partitions µ = ( µ , . . . , µ m ) and ˜ µ = (˜ µ , . . . , ˜ µ n ) they denote by N µ, ˜ µ the number of non-negative integer matrices A with row( A ) = µ and col( A ) = ˜ µ . Given sequences ( a , . . . , a ℓ ) and ( b , . . . , b ℓ )of non-negative integers, they proved the following equality [DG06, Thm. 2]: Z U ( N ) ℓ Y j =1 Sc j ( U ) a j Sc j ( U ) b j dU = N µ, ˜ µ (1.1)as long as P ℓj =1 ja j , P ℓj =1 jb j ≤ N . Here µ and ˜ µ are the partitions with a j and b j parts of size j ,respectively.Identity (1.1) answered a question raised in [HKS +
96, SHW98], where the first two moments were com-puted. The results in [DG06] inspired the study of pseudomoments of the Riemann zeta function [CG06] andwere used in [KRRGR18] to study the variance of divisor functions in short intervals. Recently, Najnudel,Paquette and Simm studied the distribution of Sc n with n growing with N [NPS20].We give a new combinatorial proof of (1.1), which makes use of the characteristic map. This is in thespirit of Bump’s derivation [Bum04, Prop. 40.4] of the Diaconis-Shahshahani moment computation [DS94].In § { P a j , P b j } ≤ N . These traces are also the completehomogeneous symmetric polynomials h n evaluated on the eigenvalues of the matrix. For a permutation π we say that S is an invariant set for π if π ( S ) = S . Equivalently, S is a union of cyclesof π . Given a partition λ = ( λ , λ , . . . , λ ℓ ) ⊢ n , we define the following function on the symmetric group S n acting on [ n ] := { , , . . . , n } : d λ ( π ) = { ( A , . . . , A ℓ ) : ·∪ A i = [ n ] , each A i is an invariant set with | A i | = λ i } , ·∪ means disjoint union. We use the letter d here as short for divisor , as these functions are analogousto divisor functions over the integers. Given µ, ˜ µ ⊢ n , let us define N µ, ˜ µ ( n ) := 1 | S n | X π ∈ S n d µ ( π ) d ˜ µ ( π ) , Proposition 2.1.
Suppose µ, ˜ µ ⊢ n . We have N µ, ˜ µ ( n ) = N µ, ˜ µ . Proof.
By definition, given a partition λ = ( λ , . . . , λ ℓ ) ⊢ n we may express d λ ( π ) as a sum over ordered setpartitions: d λ ( π ) = X ( A ,...,A ℓ ): ·∪ A i =[ n ] | A i | = λ i α A ,...,A ℓ ( π ) (2.1)where α A ,...,A ℓ is the indicator function of permutations π ∈ S n with π ( A i ) = A i for all i . Applying (2.1)with λ = µ and multiplying by (2.1) with λ = ˜ µ we obtain d µ ( π ) d ˜ µ ( π ) = X ( A ,...,A ℓ ( µ ) ) ·∪ A i =[ n ] | A i | = µ i X ( B ,...,B ℓ (˜ µ ) ) ·∪ B i =[ n ] | B i | =˜ µ i α A ,...,A ℓ ( µ ) ( π ) α B ,...,B ℓ (˜ µ ) ( π )where ℓ ( λ ) is the number of parts in a partition. Averaging this over S n and interchanging the order ofsummation, we find N µ, ˜ µ ( n ) = 1 n ! X ( A ,...,A ℓ ( µ ) ) ·∪ A i =[ n ] | A i | = µ i X ( B ,...,B ℓ (˜ µ ) ) ·∪ B i =[ n ] | B i | =˜ µ i X π ∈ S n α A ,...,A ℓ ( µ ) ( π ) α B ,...,B ℓ (˜ µ ) ( π ) . (2.2)The inner sum in the right-hand side of (2.2) counts permutations π ∈ S n for which A i are invariant sets,as well as the B j . In particular π ( A i ∩ B j ) ⊆ A i , B j , forcing π ( A i ∩ B j ) = A i ∩ B j . Conversely, given apermutation such that π ( A i ∩ B j ) = A i ∩ B j for all i, j , it necessarily satisfies π ( A i ) = A i and π ( B j ) = B j for all i, j . Thus, the inner sum counts π s with π ( A i ∩ B j ) = A i ∩ B j . The sets { A i ∩ B j } i,j are disjointand their union is [ n ], and so such π s are determined uniquely by their restrictions to A i ∩ B j , which maybe arbitrary, proving that the inner sum is Q ni,j =1 | A i ∩ B j | !. Hence, N µ, ˜ µ ( n ) = 1 n ! X ( A ,...,A ℓ ( µ ) ) ·∪ A i =[ n ] | A i | = µ i X ( B ,...,B ℓ (˜ µ ) ) ·∪ B i =[ n ] | B i | =˜ µ i Y i,j | A i ∩ B j | ! . (2.3)Observe that the n × m matrix C = ( | A i ∩ B j | ) has row( C ) = µ and col( C ) = ˜ µ . Hence N µ, ˜ µ ( n ) = 1 n ! X C =( c i,j ) a matrixcounted by N µ, ˜ µ Y i,j c i,j ! · { [ n ] = ·∪ i,j C i,j , | C i,j | = c i,j } . The inner expression in the right-hand side is the number of ordered set partitions of [ n ] into subsets C i,j ofsize c i,j (these sets correspond to A i ∩ B j and one reconstructs A i by A i = ∪ j C i,j and similarly B j = ∪ i C i,j ).This is just the multinomial (cid:18) n ( c i,j ) : 1 ≤ i ≤ ℓ ( µ ) , ≤ j ≤ ℓ (˜ µ ) (cid:19) = n ! Q c i,j ! , so that (2.3) simplifies to N µ, ˜ µ ( n ) = X X a matrixcounted by N µ, ˜ µ N µ, ˜ µ as claimed. 2 The characteristic map
Endow S n with the uniform probability measure. The characteristic (or Frobenius) map Ch ( N ) is a linearmap from class functions on S n to class functions on U ( N ), with the property that if n ≤ N it is an isometrywith respect to the L -norm, see [Bum04, Thm. 40.1]. It may be given byCh ( N ) ( f ) = 1 n ! X π ∈ S n f ( π ) p λ ( π ) , see [Bum04, Thm. 39.1]. Here λ ( π ) is the partition associated with π , and p λ is the power sum symmetricpolynomial associated with λ , evaluated at the eigenvalues of U ∈ U ( N ). Lemma 3.1.
Suppose λ ⊢ n . We have Ch ( N ) (sgn · d λ ) = e λ , where sgn is the sign representation and e λ is the elementary symmetric polynomial associated with thepartition λ .Proof. Given π ∈ S n , we set p π = p λ ( π ) for convenience. We then have, by plugging (2.1) in the definitionof Ch ( N ) (sgn · d λ ) and interchanging order of summation,Ch ( N ) (sgn · d λ ) = 1 n ! X ( A ,...,A ℓ ( λ ) ): ·∪ A i =[ n ] | A i | = λ i X π ∈ S n π ( A i )= A i sgn( π ) p π . We claim that the inner sum is e λ . Indeed, since π is determined by the restrictions π | A i , and since p λ = Q i p λ i , we have X π ∈ S n π ( A i )= A i sgn( π ) p π = ℓ ( λ ) Y i =1 X π i ∈ S Ai sgn( π ) p π i = ℓ ( λ ) Y i =1 λ i ! e λ i , where the last equality follows from the Newton-Girard identity P π ∈ S m sgn( π ) p π /m ! = e m . To finish, notethat the number of ordered set partitions of [ n ] into ℓ ( λ ) sets of sizes λ i is exactly the binomial coefficient (cid:0) nλ ,...,λ ℓ ( λ ) (cid:1) . Here we establish (1.1). Let ( a , . . . , a ℓ ) and ( b , . . . , b ℓ ) be sequences of non-negative integers with P ℓj =1 ja j , P ℓj =1 jb j ≤ N . Let µ and ˜ µ be the partitions with a j and b j parts of size j , respectively.If P ja j = P jb j , it is easy to see that both sides of (1.1) vanish. Indeed, for the right-hand side, notethat the integrand is an homogeneous polynomial in the eigenvalues fof U , whose degree is non-zero, so itsintegral must vanish by translation-invariance of the Haar measure. On the other hand, if N µ, ˜ µ is non-zero,we must have that µ and ˜ µ sum to the same number (if A = ( a i,j ) is a matrix counted by N then both µ and ˜ µ sum to P i,j a i,j ).Now assume P ja j = P jb j = n ≤ N . As Sc j ( U ) a j Sc j ( U ) b j = e µ e ˜ µ by definition, the fact that Ch ( N ) isan isometry if n ≤ N shows, through Lemma 3.1, that the integral in (1.1) is equal to1 | S n | X π ∈ S n (sgn · d µ )( π )sgn · d ˜ µ ( π ) = 1 | S n | X π ∈ S n d µ ( π ) d ˜ µ ( π ) = N µ, ˜ µ ( n ) , and the proof is concluded by applying Proposition 2.1.3 Symmetric powers
Let TrSym n ( U ) be the trace of the n th symmetric power of U ∈ U ( N ). This is also the n th completehomogeneous symmetric polynomial h n evaluated on the eigenvalues of U . Theorem 5.1.
Let { a j } ℓj =1 , { b j } ℓj =1 be sequences of non-negative integers. We have Z U ( N ) ℓ Y j =1 (TrSym j ( U )) a j (TrSym j ( U )) b j dU = N µ, ˜ µ (5.1) as long as min { P ℓj =1 a j , P ℓj =1 b j } ≤ N . Here µ and ˜ µ are the partitions with a j and b j parts of size j ,respectively. We start with the following corollary of Lemma 3.1.
Corollary 5.2.
Suppose λ ⊢ n . We have Ch ( N ) ( d λ ) = h λ . This follows from Lemma 3.1 through the existence of an involution ι on the space of symmetric polynomi-als, with the properties Ch ( N ) (sgn · f ) = ι (Ch ( N ) ( f )) [Bum04, Thm. 39.3] and ι ( e λ ) = h λ [Bum04, Thm. 36.3].Alternatively, one may repeat the proof of Lemma 3.1 with the Newton-Girard identity P π ∈ S m p π /m ! = h m .Next we prove the following well-known identity, often proved as a consequence of the RSK correspon-dence. Lemma 5.3.
Given µ, e µ ⊢ n we have X λ ⊢ n K λ,µ K λ, e µ = N µ, ˜ µ (5.2) where K λ,µ are the Kostka numbers.Proof. We may expand e λ in the Schur basis, see [Sta99, p. 335]: e µ = X λ ⊢ n K λ ′ ,µ s λ where λ ′ is the conjugate of λ . Orthogonality of Schur functions [DG06, Eq. (22)] implies that Z U ( n ) ℓ Y j =1 Sc j ( U ) a j Sc j ( U ) b j dU = X λ ⊢ n K λ ′ ,µ K λ ′ , e µ = X λ ⊢ n K λ,µ K λ, e µ . On the other hand, this integral was shown to equal N µ, e µ .We now prove Theorem 5.1. The case P ℓj =1 ja j = P ℓj =1 jb j is treated as in the secular coefficients case.Next, assume that P ja j = P jb j = n and min { P ℓj =1 a j , P ℓj =1 b j } ≤ N . The multiset of eigenvalues ofTrSym j U consists of products of j eigenvalues of U , and so the integrand in the left-hand side of (5.1) is h µ h ˜ µ . We may expand h λ in the Schur basis, see Stanley [Sta99, Cor. 7.12.4]: h µ = X λ ⊢ n K λ,µ s λ . Orthogonality of Schur functions implies that the left-hand side of (5.1) is X λ ⊢ nℓ ( λ ) ≤ N K λ,µ K λ, ˜ µ . As K λ,µ = 0 implies ℓ ( λ ) ≤ ℓ ( µ ) [Sta99, Prop. 7.10.5], and min { ℓ ( µ ) , ℓ ( µ ′ ) } = min { P a j , P b j } ≤ N byassumption, this sum is equal to the full sum P λ ⊢ n K λ,µ K λ, ˜ µ and the proof is concluded by (5.2).A version of Theorem 5.1, with max in place of min, may also be derived from formulas for averages ofratios of characteristic polynomials [CFZ05, BG06]. 4 Acknowledgements
We thank Brian Conrey and Jon Keating for comments on an earlier version of this note and Brad Rodgersfor useful suggestions. This project has received funding from the European Research Council (ERC) underthe European Union’s Horizon 2020 research and innovation programme (grant agreement No 851318).
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Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
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