The Partition Function of Log-Gases with Multiple Odd Charges

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By ﬁrst conditioning on the number of particles of each species, the partition function for the Isocharge Grand Canonical Ensemble is built up from the partition functions of the Canonical type, revealing the former to be a generating function of the latter as a function of the fugacities of each species (roughly, the probability of the occurrence of any one particle of a given charge). In section 7, we produce analogous results for charged particles placed on the unit circle in the complex plane.

1.1 Historical context

The β-ensembles are a well-studied collection of random matrices whose eigenvalue densities take a common form, indexed by a non-negative, real parameter β. Suppose µ is a continuous probability measure on R dµ R with Radon-Nikodym derivative dx = w(x). For each β >0, consider the N-point process speciﬁed by the joint probability density ∈ 1 ρ (x ,...,x )= x x β w(x ) N 1 N Z (β)N! | j − i| i N i

2 The classical β-ensembles (with β =1, 2, 4 and w(x)= e−x /2), corresponding to Hermitian matrices with real, complex, or quaternionic Gaussian entries (respectively), were first studied in the 1920s by Wishart in multivariate statistics [23] and the 1950s by Wigner in nuclear physics [22]. In the subsequent decade, Dyson and Mehta [9] unified a previously disparate collection of random matrix models by demonstrating that the three classic β-ensembles are each variations of a single action on random Hermitian matrices (representing the three associative division algebras over R). In [7], Dumitriu and Edelman provide tridiagonal matrix models for β-ensembles of arbitrary positive β, which are then used by Ram´ırez, Rider, and Virág in [16] to obtain the asymptotic distribution of the largest eigenvalue.

For each 1 n N, deﬁne the nth correlation function by ≤ ≤ 1 Rn(x1,...,xn)= ρN (x1,...,xn,y1,...,yN−n) dy1 dyN−n. (N n)! RN−n ··· − Z It turns out that the correlation function for the classic β-ensembles takes a particularly nice algebraic form. For example, when β = 2, it can be shown using only elementary matrix operations and Fubini’s Theorem that 1 Rn(x1,...,xn)= det(K(xi, xj )1≤i,j≤n), ZN (2) where the kernel K(x, y) is a certain square integrable function R R R that can most easily be expressed in terms a family of polynomials which are orthogonal with respect× to→ the measure µ. For this reason, we say

2 the classical β = 2 ensemble is an example of a determinantal point process. The details of this derivation are given in [14]. Similarly, when β = 1 or 4, 1 Rn(x1,...,xn)= Pf(K(xi, xj )1≤i,j≤n), ZN (β)

where Pf(A)= det(A) denotes the Pfaffian of an antisymmetric matrix A, and where K(x, y) is a certain 2 2 matrix-valued function whose entries are square-integrable, and which satisfies K(x, y)T = K(y, x). p We× then say the classical β = 1 and β = 4 ensembles are examples of Pfaffian point processes. This− result was first shown for circular ensembles by Dyson in [8], then for Gaussian ensembles by Mehta in [14] and then for general weights (µ) by Mehta and Mahoux in [13], except for the case β = 1 and N odd. Finally, the last remaining case was given by Adler, Forrester, and Nagao in [1]. Of fundamental concern in the theory of random matrices is the behavior of eigenvalue statistics as N . The immediate advantage of these determinantal and Pfaffian expressions for the correlation functions→ is ∞ that these matrix kernels do not essentially increase in complexity as N grows large, since the dimensions are the matrix kernel are stable, and the entries are expressed as a sum whose asymptotics are well-understood.

1.2 Hyperpfaﬃan partition functions

Derivations of the determinantal and Pfaffian expressions of the correlation functions have been presented in numerous ways over the past several decades. Of particular note is the method of Tracy and Widom [21], who first show that the partition function is determinantal or Pfaffian, and then use matrix identities and generating functions to obtain a corresponding form for the correlation functions.

But recognizing the partition function ZN (β) as the determinant or Pfaffian of a matrix of integrals of appropriately chosen orthogonal polynomials is essential and nontrivial. One way to do this is to apply the Andreif determinant identity [2] to the iterated integral which defines ZN (β). This is immediate when β = 2, and viewing the Pfaffian as the square root of a determinant, this identity can also be applied (with some additional finesse) when β = 1 or 4. However, viewing the Pfaffian in the context of the exterior algebra allows us to extend the Andreif determinant identity to analogous Pfaffian identities (referred to as the de Bruijn integral identities [6]).

In 2002, Luque and Thibon [12] used techniques in the shuffle algebra to show that when β = L2 is an even square integer, the partition function ZN (β) can be written as a Hyperpfaffian of an L-form whose coefficients are integrals of Wronskians of suitable polynomials. Then in 2011, Sinclair [19] used other combinatorial methods to show that the result also holds when β = L2 is an odd square integer.

In this volume, we consider multicomponent ensembles in which the joint probability density functions generally have the form 1 ρ (x ,...,x )= x x βf(i,j) w (x ), N 1 N Z (β)N! | j − i| i i f i

In 2012, Sinclair showed (using his same methods) the partition function Zf (β) has a Berezin integral expression when each √βLj is an even integer. Herein, we extend to arbitrary positive integers √βLj Z>0 using shuﬄe algebra techniques, analogous to the methods of Thibon and Luque. Note, single-component∈

3 β-ensembles are a subset of multicomponent ensembles in which f(i, j) = 1, and the Berezin integral is a generalization of the Hyperpfaﬃan. Thus, one consequence of this work is a new, streamlined derivation of the Hyperpfaﬃan partition functions for the single-component β-ensembles in which β = L2 is any square integer, even or odd.

Though multicomponent ensembles are the focus of this volume, our methods further generalize to other ensembles. We explore many of these possibilities in a forthcoming volume. One key observation is that ρN can be written as a determinant (typically easier than expressing ZN as a determinant). More generally, we can replace the partition function ZN (or Zf ) with an iterated integral of any determinant ﬁtting certain criteria. In subsection 3.1, we describe how to modify the statement of the main results of this volume, producing a vast generalization of the de Bruijn integral identities.

1.3 The multicomponent setup

Let J Z>0 be a positive integer, the maximum number of distinct charges in the system. Let L~ = ∈ J (L1,L2,...,LJ ) (Z>0) be a vector of distinct positive integers which we will call the charge vector of the ∈ J system. Let M~ (Z≥ ) be a vector of non-negative integers which we will call the population vector of the ∈ 0 system. Each Mj gives the number (possibly zero) of indistinguishable particles of charge Lj . Let x = (x1, x2,..., xJ ) RM1 RM2 RMJ ∈ × ×···× so that xj = (xj , xj ,...,xj ) RMj for each j. We call x the location vector of the system in which each 1 2 Mj ∈ j R j xm gives the location of a particle of charge Lj . We call x the location vector for the species with ∈ j charge Lj. If some Mj = 0, then we take x to be the empty vector.

The particles are assumed to interact logarithmically on an inﬁnite wire so that the contribution of j k energy to the system by two particles of charge Lj and Lk at locations xm and xn respectively is given by k j LjLk log xn xm . If U is the potential on the system, then at inverse temperature β, the total potential −energy of the| system− | is given by

J Mj J Mj Mk E (x)= β L U(xj ) β L2 log xj xj β L L log xk xj . M~ j m − j | n − m|− j k | n − m| j=1 m=1 j=1 m

With this setup, the relative density of states (corresponding to varying location vectors x) is given by the Boltzmann factor Ω (x) = exp( E (x)) M~ − M~ J Mj J Mj Mk 2 βL βLj L = exp βL U(xj ) xj xj j xk xj k . − j m × n − m × n − m j=1 m=1 j=1 m

Later, it will be convenient to write WM~ (x) in place of the ﬁrst of the three iterated products above. In the case when √βL Z for all j, we will also write det V L~ ,M~ (x) in place of the product of the remaining two j ∈ L~ ,M~ iterated products above. In subsection 4.4, we construct the ma trix V (x) of which this is the determinant. Then the probability of ﬁnding the system in a state corresponding to a location vector x is given by the joint probability density function W (x) det V L~ ,M~ (x) Ω ~ (x) M~ ρ (x)= M = , M~ ZM~ M1!M2! MJ ! ZM~ M1 !M2! MJ ! ··· ··· 4 where the partition function (of the Canonical Ensemble) ZM~ is the normalization constant given by

1 M1 1 M2 2 MJ J ZM~ = ΩM~ (x) dν (x ) dν (x ) dν (x ) M !M ! M ! RM1 ··· RMJ ··· 1 2 ··· J Z Z 1 L,~ M~ M1 1 M2 2 MJ J = det V (x) dµ1 (x ) dµ2 (x ) dµJ (x ), M1!M2! MJ ! RM1 ··· RMJ ··· ··· Z Z

with Lebesgue measure νMj on RMj and dµ (x)= w (x) dx = exp( βL U(x)) dx. j j − j Note, the factorial denominators appear since particles of the same charge are indistinguishable, giving many different representatives for each state. In particular, the integrand is invariant under permutation of xj , xj ,...,xj for any j fixed. At this point, it is necessary to assume the potential U is one for which { 1 2 Mj } ′ 2 ′ L~ ,M~ ZM~ is finite. Also, replacing β with β = β/b and replacing each Lj with Lj = bLj leaves det V (x) unchanged. Then replacing U with U ′ = bU leaves W (x) unchanged. Thus, for computational purposes, M~ we can always assume β = 1.

Next, allowing the number of particles of each species to vary, let P (M~ ) be the probability of ﬁnding the system with population vector M~ . Let ~z = (z ,...,z ) (R )J be a vector of positive real numbers called 1 J ∈ >0 the fugacity vector. Classically, the probability P (M~ ) is given by

Z ~ ~ M1 M2 MJ M P (M)= z1 z2 zJ , ··· ZN

where ZN is the partition function of the Isocharge Grand Canonical Ensemble (corresponding to ﬁxed total charge N) given by Z = zM1 zM2 zMJ Z . N 1 2 ··· J M~ L~ ·XM~ =N In the above expression, the vector L~ of allowed charges is ﬁxed, so we are summing over allowed population ~ J vectors M. A population vector is valid only when the sum of the charges j=1 Lj Mj is equal to the prescribed total charge N. P

This ZN is the primary object of interest to us. In addition to the obvious dependence on charge vector L~ and total charge N, this expression also varies with potential U and inverse temperature β. The potential U dictates the external forces experienced by each particle individually, affecting the measures against which we are integrating. The inverse temperature β influences the strength of the interactions between the particles, affecting the exponents on the interaction terms in the Boltzmann factor. Taking the fugacity vector ~z to be a vector of indeterminants, ZN is a polynomial in these indeterminants which generates the partition functions of the Canonical Ensembles.

Recall, each ZM~ is an iterated integral in many variables. Our goal here is not to compute these integrals for any particular choice of several parameters. Instead, we will show, in general, how to reduce the diﬃculty of the computation to (sums of products of) only single or double integrals. Sinclair (2012) showed for certain values of β (for which √βLj Z for all j), ZN can be expressed as the Berezin integral (with respect to the volume form on RN ) of the exponential∈ of an explicit element (alternating tensor, also form) in the N exterior algebra (R ) in the case when at most one of the Lj is odd. In the case when exactly one of the Lj is odd, say L1, this required the additional restriction that N be even, which would force M1 to be even. V We will show his expression can be extended to arbitrary L~ (for which any number of the Lj may be odd), and in the case when N is odd, we give an analogous Berezin integral expression with respect to the volume form on RN+1.

5 2 Preliminary deﬁnitions

In this section, we introduce a mix of conventions and deﬁnitions which simplify the statement of our main results. First, for any positive integer N, let N denote the set 1,...,N . Assuming positive integers K N, let t : K N denote a strictly increasing function from K{to N, meaning} ≤ ր 1 t(1) < t(2) < < t(K) N. ≤ ··· ≤ It will be convenient to use these increasing functions to track indices used in denoting minors of matrices and elements of exterior algebras, among other things (often in place of, but sometimes in conjunction with, permutations). For example, given an N N matrix V , Vt might denote the K K minor composed of the rows t(1),..., t(K), taken from the ﬁrst× K columns of V . More conventions× related to indexing and permutations (which are relevant to the proofs) are introduced in section 4 but are not necessary for the statement of our main results.

2.1 Wronskians

For any non-negative integer l, define the lth modified differential operator Dl by 1 dlf Dlf(x)= , l! dxl 0 ~ ~ L with D f(x) = f(x). Define the modified Wronskian, Wr(f, x), of a family, f = fn n=1, of L many sufficiently differentiable functions by { }

~ l−1 L Wr(f, x) = det D fn(x) n,l=1 . We call this the modified Wronskian because it differs from the typical Wronskian (used in the study of elementary differential equations to test for linear dependence of solutions) by a combinatorial factor of L l=1 l!.

Q N A complete N-family of monic polynomials is a collection ~p = pn n=1 such that each pn is monic of L { } degree n 1. Given t : L N, denote ~pt = pt . Then the (modiﬁed) Wronskian of ~pt is given by − ր { (k)}k=1 l−1 L t t Wr(~p , x) = det D p (k)(x) k,l=1 . 2.2 The Berezin integral

N K N Let ε ,...,ε be a basis for R . For any injection t : K N, let εt (R ) denote 1 N → ∈ εt = εt εt εt . V (1) ∧ (2) ∧···∧ (K) K N N N Then εt t : K N is a basis for (R ). In particular, (R ) is a one-dimensional subspace we call the determinantal{ | ր line}, spanned by V V ε = ε = ε ε ε , vol id 1 ∧ 2 ∧···∧ N which we call the volume form (in RN ). For each 0

6 ∂εt and then extend linearly. If n t(K), meaning εn appears as a factor in εt, then is the result of permuting ∈ ∂εn εn to the front and then removing it, picking up a sign associated with changing the order in which the basis ∂εt elements occur. If εt does not have ε as a factor, then = 0. Given an injection s : L N, we deﬁne n ∂εn N N → the Berezin integral [3] (with respect to εs) as a linear operator (R ) (R ) given by → ∂V ∂V ∂ εt dεs = εt dεs(1) dεs(2) dεs(L) = εt. ··· ∂εs ··· ∂εs ∂εs Z Z (L) (2) (1) N N Our main results are stated in terms of Berezin integrals with respect to the volume form εvol (R ). K N ∈ Note, if εt (R ) for any K

2.3 Exponentials of forms

For ω (RN ) and positive integer m, we write ∈ V ω∧m = ω ω, ∧···∧ with ω appearing as a factor m times. By convention, ω∧0 = 1. We then deﬁne the exponential ∞ ω∧m exp(ω)= . m! m=0 X Moreover, suppose ω = ω + ω + + ω where each ω Lj (RN ) and each L even, then (we say each 1 2 ··· J j ∈ j ωj is a homogeneous even form of length Lj and) it is easily veriﬁed V exp(ω) = exp(ω + + ω ) = exp(ω ) exp(ω ). 1 ··· J 1 ∧···∧ J

In our main results, the forms we are exponentiating are non-homogeneous. However, we get a homogeneous form in the case when we only have one species of particle. In that case, exactly one summand in the exponential will live at the determinantal line. Assuming ω L(RN ) with LM = N, we get ∈ ∞ ω∧m Vω∧M exp(ω) dε = dε = dε = PF(ω), vol m! vol M! vol m=0 Z Z X Z ∧ ω M where PF(ω) is the Hyperpfaffian of ω, the real number coefficient on εvol in M! . Thus, this Berezin integral is the appropriate generalization of the Hyperpfaffian. To avoid confusing this Berezin integral with other integrals which appear in our computations, we’ll write

BEvol(ω)= exp(ω) dεvol, Z where the subscript on the left hand side indicates which form we are integrating with respect to.

The partition function of an ensemble with a single species has been shown to have a Hyperpfaﬃan expression (for certain β) [19]. As we generalize to ensembles with multiple species (and therefore non- homogeneous forms), we replace the Hyperpfaﬃan with the more general Berezin integral (of an exponential).

7 3 Statement of results

Recall the setup from subsection 1.3. Let ~p be a complete N-family of monic polynomials. Deﬁne

γj = Wr(~pt, x) dµj (x) εt, R t ր :LXj N Z and deﬁne η = Wr(~pt, x)Wr(~ps,y) dµ (x)dµ (y) εt εs. j,k j k ∧ t ր s ր x

Theorem 3.1. If all Lj are even, then

J Z = BE z γ . N vol  j j  j=1 X  

The above theorem is Sinclair’s (2012) for which we give a diﬀerent proof and then the following generalization:

Theorem 3.2. If the ﬁrst r many Lj are even and N is even, then

r J J Z = BE z γ + z z η . N vol  j j j k j,k j=1 j=r+1 X X k=Xr+1   Theorem 3.3. If the ﬁrst r many Lj are even and N is odd, then

r J J J Z = BE z γ + z z η + z γ ε , N vol1  j j j k j,k j j ∧ N+1 j=1 j=r+1 j=r+1 X X k=Xr+1 X   where BE includes the Berezin integral with respect to ε = ε ε N+1(RN+1). vol1 vol1 vol ∧ N+1 ∈ V

For our methods, it is necessary to extend the basis by εN+1 so that the new volume form εvol1 has even length N + 1. More generally, we can write

ε = ε ξ = ε ε ε ε . volk vol ∧ k vol ∧ N+1 ∧ N+2 ∧···∧ N+k Then for any ω (RN ) (RN+k), we have ∈ ≤ V V ω dε = ω ε ε dε dε dε = ω ξ dε . vol ∧ N+1 ∧···∧ N+k vol N+1 ··· N+k ∧ k volk Z Z Z Thus, we can embed any Berezin integral computation in a higher dimension as desired. Corollary. In the single species case (N indistinguishable particles of charge L), we get the known Hyperp- faﬃan expression (Sinclair 2011):

ZN = BEvolk (ω) = PF(ω), where ω and k depends on N and L.

1. If L is even, then ω = γ1 and BEvolk = BEvol.

8 2. If L is odd and N is even, then ω = η1,1 and BEvolk = BEvol. 3. If L is odd and N is odd, then ω = η + γ ξ and BE = BE . 1,1 1 ∧ L volk volL

Note, we extend by ξL instead of just ξ1 = εN+1 in case 3 only so that γ1 ξL is a 2L-form and therefore ω is homogeneous. Every choice of k produces a diﬀerent but equally valid Berezin∧ integral expression. We obtain the (Pfaﬃan) de Bruijn integral identities for classical β = 1 and β = 4 when L = 1 and L = 2, respectively.

To prove these theorems, we start by giving general methods (in the shuffle algebra) for manipulating iterated integrals of determinantal integrands in section 4. The conclusion of these methods is a generalization of the de Bruijn integral identities, given in subsection 3.1. In section 5, we apply these identities first to ~ ZM~ , the partition function of the Canonical Ensemble with arbitrary but fixed population M. In section 6, we sum over all possible population vectors M~ to obtain ZN , the partition function of the Isocharge Grand Canonical Ensemble.

3.1 Generalized de Bruijn identities

Let N = L + +L . Deﬁne K = j L . Let B(~x) be an N N matrix whose entries are single variable 1 ··· J j k=1 k × integrable functions of variables ~x = (x1,...,xJ ). Explicitly, the ﬁrst L1 many columns are functions of x1, P the second L many columns are functions of x , and so on up through x . For t : L N, let Bt(x ) 2 2 J j ր j denote the L L minor of B(~x) given by j × j

Lj t t B (xj )= B(~x) (l),n+Kj−1 l,n=1 , equivalently obtained from B(~x) by taking the rows t(1),..., t(Lj) from the Lj many columns in the same variable xj . Deﬁne B γj = det Bt(xj ) dxj εt, R t ր :LXj N Z and deﬁne B η = det Bt(x ) det Bs(x ) dx dx εt εs. j,k j · k j k ∧ t ր s ր xj

Theorem 3.4. Suppose the ﬁrst r many Lj are even, then

det B(~x) dx dx = ω dε , 1 ··· J vol Z−∞

1. If N is even, then r (J−r)/2 1 ω = γB ηB . J−r j ∧ r+2m−1,r+2m r + 2 ! j=1 m=1 ^ ^ 2. If N is odd, then

r (J−r−1)/2 1 ω = γB ηB γB. J−r−1 j ∧ r+2m−1,r+2m ∧ J r +1+ 2 ! j=1 m=1 ^ ^ 9 B B For 1 j r, Lj is even, and γj is an even Lj-form. For the Lj which are odd, ηj,k combines minors of odd L ≤ L≤ dimensions with minors of odd L L dimensions to produce an even (L + L )-form. In j × j k × k j k case 1, the requirement that N be even means there are an even number of odd Lj to be paired down into (J r)/2 pairs. In case 2, there are an odd number of odd L , so γB remains as an odd L -form. − j J J

Compared to Theorems 3.2 and 3.3, det B(~x) takes the place of the Boltzmann factor ΩM~ (x), so that det Bt(xj ) and det Bs(xk) take the places of Wr(~pt, x) and Wr(~ps,y), respectively. Recall (from subsection 1.3), the partition function ZN of the Isocharge Grand Canonical Ensemble is a linear com- bination of the partition functions ZM~ of the Canonical Ensembles, with the sum taken over all possible ~ ~ populations M. For each M, we get a diﬀerent Boltzmann factor integrand ΩM~ (x). Moreover, the deﬁning integral is taken over all of RN . The wedge products which appear in Theorem 3.4 should be thought of as just one of the summands which appears in the expansion of the exponentials which appear in Theorems 3.2 and 3.3.

Note, we assume the functions which make up B(~x) are suitably integrable so that all integrals which B B appear in γj and ηj,k are ﬁnite. However, we do not assume any resemblance between the Lj many columns in xj and the Lk many columns in xk. Assuming some additional consistency, we obtain a Hyperpfaﬃan analogue of the de Bruijn integral identities. B Corollary. Suppose L1 = = LJ = L. Under the additional assumption that γj = γ for all j, and B ··· ηj,k = η for all j, k (typically because the entries of B(~x) in one variable xj are the same as the entries in any other variable xk),

det B(~x) dx dx = BE (ω) = PF(ω), 1 ··· J volk Z−∞

1. If L is even, then ω = γ and BEvolk = BEvol.

2. If L is odd and M is even, then ω = η and BEvolk = BEvol. 3. If L is odd and M is odd, then ω = η + γ ξ and BE = BE . ∧ L volk volL

Further generalization in Theorem 3.4 is still possible if desired. Suppose instead the ﬁrst L1 columns of B(~x) are made up of functions, not necessarily single variable, of variables x1,...,xa, and the next L2 B columns of B(~x) are made up of functions of variables xa+1,...,xb. If L1 is even, then we replace γ1 with

B γ = det Bt(x ,...,x ) dx dx εt, 1 1 a 1 ··· a t ր −∞

B η = det Bt(x ,...,x ) det Bs(x ,...,x ) dx dx εt εs. 1,2 1 a · a+1 b 1 ··· b ∧ t ր s ր −∞

4 Algebraic lemmas

For any injection t : K N, let Qt denote Qt(1),...,t(K) whenever it is clear from context Q admits K many → K indices, and let Qt denote Qt whenever it is clear from context Q admits only one index. For any { (k)}k=1

10 permutation σ SK , we can view σ : K K as a bijection and then write Qt◦σ to denote Qt◦σ(1),...,t◦σ(K) K ∈ → or Qt as appropriate in context. { ◦σ(k)}k=1

For example, when V is a matrix, Vt is a minor. We should think of Vt as a single object with K many indices (which indicate a choice of K many rows t(1),..., t(K) from which our minor is constructed). Simi- K N larly, if ω (R ), then At might denote the coefficient of εt (equivalently, an entry in a K-dimensional ∈ N K hyper array). In contrast, if f~ = f is a family of functions, then f~t = ft is a subfamily of V k k=1 (k) k=1 K functions, each indexed by a single{ } integer. In our statement of our main results,{ } we use these increasing function subscripts in both ways, but it is clear from context how these subscripts should be applied differently to different objects.

Let t : K N denote a strictly increasing function from K to N. Note, every injection s : K N ր → can be written uniquely as s = t σ for some t : K N and σ SK . For any t : K N, there exists a unique complementary t′ : N K◦ N with t(K) րt′(N K) =∈ N. Deﬁne sgn(t) toր be the signature of the permutation σ S given− by ր ∪ − ∈ N t(k) if k K σ(k)= ∈ . t′(k K) if k N K ( − ∈ \ Equivalently,

sgn(t)= εt εt′ dε . ∧ vol Z For any Λ = (λ ,...,λ ) which partitions N and t : N M, write t = (t ... t ) to indicate a decompo- 1 K → 1| | K sition of t in which t1 is the restriction of t to the ﬁrst λ1 positive integers, and each tk is the restriction of t to the next λk positive integers. For convenience, we will treat each tk as having domain λk instead of the appropriate subset of N of size λk.

Conversely, for any t : λ M,..., t : λ M, we can construct (t ... t )= t : N M by deﬁning 1 1 → K K → 1| | K → for each n N, t(n) = t n k−1 λ where this k, which depends on n, is the largest k for which the ∈ k − j=1 j diﬀerence inside the parentheses is positive. As before, it will be convenient to identify the restrictions of P this new t with the original t1,..., tK even though the domains aren’t exactly the same.

In general, this is just a bookkeeping device which gives us a convenient notation for a choice of indices from the codomain M. For example, when σ : N N is a permutation, we can think of σ as sorting N → many possible indices into K blocks of diﬀerent sizes λ1,...,λK as speciﬁed by the images of the σk.

4.1 Decomposition of the symmetric group

For any Λ = (λ ,...,λ ) which partitions N, let H(Λ) S denote the Young subgroup, meaning H(Λ) = 1 K ⊆ N ∼ Sλ1 SλK . Viewing σ as a function from N to N, we can write the decomposition (with respect to Λ) as σ×···×= (σ ... σ ). Then we should think of these σ belonging to the appropriate S . 1| | K k λk

Let Sh(Λ) SN denote the subset of shuﬄe permutations. These are permutations which satisfy σ(i) < σ(j) whenever⊆ λ + + λ

11 Let Sh◦(Λ) Sh(Λ) denote the subset of ordered shuﬄe permutations. These are shuﬄe permutations which also satisfy⊆

σ(1) < σ(λ + 1) < σ(λ + λ + 1) < < σ(λ + + λ − + 1). 1 1 2 ··· 1 ··· K 1 Using the decomposition σ = (σ ... σ ), we can conveniently rewrite the above condition as 1| | K σ (1) < σ (1) < < σ (1). 1 2 ··· K Let Bl(Λ) S denote the subset of block permutations. A block permutation represents shuffling a deck ⊆ N of cards by first separating the deck into a pile of the first λ1 cards, a pile of the second λ2 cards, and so on, then reassembling the deck without interlacing the piles or shuffling within any of the piles. Using the decomposition σ = (σ1 ... σK ), block permutations are permutations for which each σk : λk N acts by σ (j) = (j 1)+ σ (1).| | → k − k Clearly, any block permutation is determined entirely by the action on the first element of each block (of λ elements), of which there are K many. Given a block permutation σ, define θ S to be the k σ ∈ K unique permutation for which θσ(i) <θσ(j) if and only if σi(1) < σj (1). Heuristically, σ moves blocks of λk consecutive elements together. θσ is the unique action on the K many blocks determined by σ. The map σ θ is bijective, so Bl(Λ) = S . 7→ σ ∼ K

Note, because the blocks have (possibly) diﬀerent sizes λj , block permutations do not (in general) preserve partitions. Deﬁne σ Λ = (λ −1 ,...,λ −1 ), θσ (1) θσ (K) obtained from Λ by reordering its entries according to θσ.

In the sequel, we make use of the following Lemma which decomposes permutations in the symmetric group as products of ordered shuﬄes, block permutations, and young permutations (composed in the opposite order).

Lemma 4.1. Let Λ = (λ1,...,λK ) be a partition of N. Given any ϕ SN , there exists unique permutations τ H(Λ), π Bl(Λ), and σ Sh◦(Λπ) so that ϕ = σ π τ. ∈ ∈ ∈ ∈ ◦ ◦

For completeness, we include a proof of this lemma in Appendix A.

4.2 Chen’s lemma

For a set X and a ring R, let R X denote the free unital algebra on X over R. Given u = u1 uk and ′ h i ··· u = u u , deﬁne an operation ¡ on R X as follows: k+1 ··· n h i ′ − − u ¡ u = u 1 u 1 σ (1) ··· σ (n)

σ∈Sh(Xk,n−k) ¡

and by e e = e for the empty word e R X . Denote by R X ¡ the algebra R X with (shuffle) ∈ h i h i h i multiplication ¡, which is called the Shuffle Algebra on X. The shuffle product was first introduced by Eilenberg and Mac Lane in [10].

Let be the Hilbert space L2(R) of square integrable functions with respect to Lebesgue measure on R, and supposeH H is a ﬁnite-dimensional subspace of with basis X. We assume H is large enough to contain all functions of interest to us. H

12 k For any σ S , let ∆ (σ) (a,b) denote the region where a < x −1 < < x −1 < b. For each ∈ k k ⊂ σ (1) ··· σ (k) non-negative integer k, deﬁne a linear functional on the kth graded component of R X = T (V ) by h·ik h i ∼ f f = f (x ) f (x ) dx dx . h 1 ⊗···⊗ kik 1 1 ··· K k 1 ··· k Z∆k(id) Note, T (V ) denotes the tensor algebra to which R X is isomorphic. Though not strictly necessary, we write f f T (V ) instead of f f R X to avoidh i confusing concatenation with function multiplication. 1⊗···⊗ k ∈ 1 ··· k ∈ h i ∞ R th This collection k k=0 deﬁnes a functional on X whereby acts as k on the k graded component of a non-homogeneous{h·i } tensor. The followingh·i lemma,h i due to Chenh·i [5], assertsh·i that this operator R

is actually an algebra homomorphism from T (V )¡ to : h·i

Lemma 4.2 (Chen). If f,g R X , then f ¡ g = f g . ∈ h i h i h ih i

A major hurdle in computing the partition function ZM~ is the absolute value inside the integral. We will remove the absolute value by decomposing the domain of integration into these totally ordered subsets. How- ever, when we change the domain of integration, we lose the ability to use Fubini’s Theorem. Chen’s Lemma serves the role of Fubini’s Theorem, provided we can demonstrate the integrand to have the appropriate form.

Proof. We can assume f = f f is a pure tensor of length k and g = f f is a pure tensor

1 ⊗···⊗ k k+1 ⊗···⊗ n

¡ ¡ of length n k (because is linear and ¡ distributes over addition). Then f g = f g is an n-fold − h·i h i h in iterated integral over ∆ (id). Likewise, f g = f g − is the product of a k-fold iterated integral over n h ih i h ikh in k ∆ (id) and an (n k)-fold iterated integral over ∆ − (id). Under the integrability criteria on X, Fubini’s k − n k Theorem tells us this is the same as an n-fold iterated integral over the product space ∆ (id) ∆ − (id). k × n k Key to the proof of Chen’s Lemma is the observation that

∆ (id) ∆ − (id) = Z ∆ (σ), k × n k ∪ n σ∈Sh([k,n−k) where Z (a,b)n has measure 0. To this end, let Y be the set of points in (a,b)n whose coordinates are ⊂ n all distinct, and then let Z = ((a,b) Y ) (∆k(id) ∆n−k(id)). If ~x (∆k(id) ∆n−k(id)) Z Y , then its coordinates are all distinct and\ can be∩ ordered× according to some∈ permutation,× meaning ~x\ ∆⊂(σ) ∈ n for a unique σ Sn. Since ~x ∆k(id) ∆n−k(id), this permutation is one which separately preserves the relative order of∈ the ﬁrst k coordinates∈ × and the relative order of the remaining n k coordinates, meaning σ Sh(k,n k). Thus, − ∈ − f g = f (x ) f (x ) dx dx h ih i 1 1 ··· n n 1 ··· n Z∆k(id)×∆n−k(id)

= f1(x1) fn(xn) dx1 dxn. ∆n(σ) ··· ··· σ∈Sh(Xk,n−k) Z

By relabeling the variables as xj = yσ(j), we can rewrite this sum as

= f1(yσ(1)) fn(yσ(n)) dy1 dyn ∆n(id) ··· ··· σ∈Sh(Xk,n−k) Z

= fσ−1(1)(y1) fσ−1(n)(yn) dy1 dyn ∆n(id) ··· ··· σ∈Sh(Xk,n−k) Z

= f ¡ g . h i

13 Until section 7, we will only need (a,b) = ( , )= R, but we have shown Chen’s Lemma to hold for more general intervals. Also, we do not need our−∞ functions∞ to be real-valued. We just need the codomain to be associative and commutative, provided our functions are appropriately integrable. In particular, we will use complex-valued functions on real domain [0, 2π) for the circular ensembles in section 7.

4.3 Exterior shuﬄe algebra

j Let Qt j J, t : Lj N be a subset of an alphabet I, and let F be a field of characteristic 0. We use single{ | integer∈ superscripts→ } to emphasize there are different Q’s which admit different numbers of integer j subscripts, chosen by t’s of different sizes. For each t : L N, define At F I by j → ∈ h i j j At = sgn(τ)Qt◦τ . τ∈S XLj j We call this the antisymmetrization of Qt . We should think of this as analogous to taking a pure tensor in the tensor algebra and then adding to it all possible orderings of the basis vectors with signs. Let V be a

rank N free module over R = F I ¡ . Deﬁne α V by h i j ∈ R Vj j αj = Qt εt = At εt. t → t ր :LXj N :LXj N

We call this an antisymmetrized Lj -form.

Let M~ = (M ,...,M ), and let L~ = (L ,...,L ) such that each L is even. Let N = L~ M~ , and 1 J 1 J j · let Λ = (L1,...,L1,L2,...,L2,...,LJ ,...,LJ ) = (λ1, λ2,...,λK ) with each Lj appearing Mj times. Let K = J M . For σ S , write σ = (σ σ ) so that each σ is the restriction of σ : N N to a j=1 j ∈ N 1| · · · | K k → subset of size λk = Lj of which there are Mj many. In particular, when σ Sh(Λ), we have σk : Lj N. P ∈ ր

Lemma 4.3. Under the above assumptions (particularly requiring each Lj to be even) and deﬁnitions (such j

as how to obtain αj from the Qt ), we have ¡

∧¡ M1 ∧ MJ ¡ α ¡ α sgn(σ)Q1 QJ = 1 ∧ ···∧ J dε . σ1 ··· σK K! vol ∈

σXSN Z ¡ In particular, when J =1, the right hand side is PF¡ (α1), where the subscript is added to emphasize the

j ¡

coeﬃcients At are in F I ¡ in which multiplication of coeﬃcients is done by . h i

j Though this lemma holds for a more general collection of Qt , we should think of the left hand side as j j being an N N determinant. A single Qσk is a product of Lj many entries from the matrix. Aσk , the × j antisymmetrization of Qσk , is the determinant of an Lj Lj minor selected by σk. In summary, this Lemma 4.3 transforms any determinantal integrand into one for× which Chen’s Lemma will apply. This is functionally similar to the Laplace expansion of the determinant (over complimentary Lj Lj minors) which Sinclair uses in his proofs. ×

j Proof. Starting with the right hand side, note each factor αj is a sum of At εt’s. If we expand the product j of the sums, each summand will be a product of some At εt’s. Any time t : L N and s : L N have j ր k ր overlapping ranges, εt εs = 0. Thus, each nonzero summand corresponds to a permutation (t1 ... tK )= σ S . Since each t ∧is an increasing function, we have σ Sh(Λ). We use the Berezin integral| | dε ∈ N k ∈ vol R

1 J ¡ t t ¡ because it sends εσ = ε 1 ε K to sgn(σ) and picks up the coeﬃcients Aσ1 AσK . We can rewrite the right hand side as ∧···∧ ···

1 1 J ¡ RHS = sgn(σ)A ¡ A . K! σ1 ··· σK σ∈XSh(Λ) We eliminate the factorial denominator by requiring σ1(1) < σ2(1) <...<σK (1).

1 J ¡ RHS = sgn(σ)Aσ1 ¡ AσK . ◦ ··· σ∈ShX(Λ) j Expanding each Aσk according to the deﬁnition, we get

1 J ¡ RHS = sgn(σ) sgn(τ1)Q ¡ sgn(τK )Q .  σ1◦τ1  ···  σK ◦τK  ∈ ◦ τ ∈S τ ∈S σ XSh (Λ) 1XL1 KXLJ     Collect the τ ’s as a single element of H(Λ) (S )M1 (S )MJ with sgn(τ) = sgn(τ ) sgn(τ ), k ∼= L1 LJ 1 K so that ×···× ···

1 J ¡ RHS = sgn(σ) sgn(τ)Qσ1◦τ ¡ QσK ◦τ . ◦ ··· σ∈ShX(Λ) τ∈XH(Λ)

Next, we apply an identity of the ¡ operation. Note, we are shuﬄing individual letters together, not strings of letters, so the sum is over π SK = Sh(1,..., 1). The action on the subscripts can also be viewed as a permutation of the K many “blocks”∈ of N as prescribed by the partition Λ. Thus,

RHS = sgn(σ) sgn(τ) Q1 QJ σπ(1) ◦τ ··· σπ(K)◦τ ◦ ∈ σ∈ShX(Λ) τ∈XH(Λ) πXSK = sgn(ϕ)Q1 QJ . ϕ1 ··· ϕK ∈ ϕXSN Note, the equality in the last line is not an obvious one but follows from Lemma 4.1. In the context of Lemma 4.3, all of the block sizes Lj are even, so sgn(π) = 1 for all π Bl(Λ). Thus, sgn(ϕ) = sgn(σ π τ) = sgn(σ)sgn(τ). ∈ ◦ ◦

4.4 Conﬂuent Vandermonde determinant

In this subsection, we demonstrate that the joint probability density function can be written as a determinant (to which we can apply Lemma 4.3). Fix charge vector L~ , population vector M~ , and location vector x as in subsection 1.3. Recall N = J M L . Let f~ = f N be a family of max(L ,...,L ) 1 times j=1 j j { n}n=1 1 J − differentiable functions. For each j, define the N Lj matrix P × N,L Lj l−1 j V (x)= D fn(x) n,l=1 . For each xj RMj , define the N M L matrix ∈ × j j V Lj ,Mj (xj )= V Lj (xj ) V Lj (xj ) V Lj (xj ) . 1 2 ··· Mj h i Finally, define the N N matrix × ~ ~ V L,M (x)= V L1,M1 (x1) V L2,M2 (x2) V LJ ,MJ (xJ ) , ··· 15 j ~ in which each variable xm appears in Lj many consecutive columns, generated from f by taking derivatives. The Wronskians which appear in section 3 are merely the determinants of the univariate Lj Lj minors of this matrix. ×

With the additional restriction that f~ be a complete N-family of monic polynomials, we call V L,~ M~ (x) the conﬂuent Vandermonde matrix (with respect to shape L,~ M~ ) in variables x. Under these conditions (on f~), it is well known [15] that

J Mj Mk 2 ~ ~ L Lj Lk det V L,M (x)= xj xj j xk xj . n − m × n − m j=1 m

L,~ M~ ΩM~ (x)= WM~ (x) det V (x) .

Moreover, if we deﬁne the N L matrices × j N,L HLj (x) = exp( U(x))V Lj (x)= exp( U(x)) Dl−1f (x) j − − · n n,l=1 and the combined N N matrix HL~ ,M~ (x), then ×

J Mj ~ ~ ~ ~ ~ ~ det HL,M (x) = exp L U(xj ) det V L,M (x) = W (x) det V L,M (x) . − j m M~ j=1 m=1 Y Y

L,~ M~ L,~ M~ Thus, integration of V (x) with respect to the dµj ’s is equivalent to integration of H (x) with respect to Lebesgue measure. For us to use Lemma 4.3, it is important the entire integrand ΩM~ (x) be determinantal, L,~ M~ with the extra weight functions WM~ (x) incorporated into H (x).

As an example, consider one charge 2 particle, one charge 3 particle, and three charge 1 particles with potential U(x) = x2. This gives us L~ = (2, 3, 1), M~ = (1, 1, 3), and N = 8. For simplicity, we will use the ~ n−1 N variables x = (a,b,c1,c2,c3). Let f = x n=1. Then the three columns corresponding to the charge 3 particle are { } 1 0 0 b 1 0 b2 2b 1  3 V (b)= b3 3b2 3b  .    . .   . .    b7 7b6 21b5   In the third column, we have not just the second derivative but also a denominator of 2!. One consequence of these l! denominators in Dl−1 is that we get 1’s on this diagonal. We also get three columns corresponding to the three charge 1 particles 1 1 1 c1 c2 c3   1,3 c2 c2 c2 V (c1,c2,c3)= 1 2 3 .  . .   . .    c7 c7 c7  1 2 3  

16 Together, the full 8 8 conﬂuent Vandermonde matrix is × 1010 0 111 a 1 b 1 0 c1 c2 c3 a2 2a b2 2b 1 c2 c2 c2 L~ ,M~ 1 2 3 V (x)= a3 3a2 b3 3b2 3b c3 c3 c3 .  1 2 3  . .   . .    a7 7a6 b7 7b6 21b5 c7 c7 c7  1 2 3   We obtain HL~ ,M~ (x) by multiplying the ﬁrst two columns by exp( a2), the next three columns by exp( b2), and the last three columns by the appropriate exp( c2). This changes− the determinant by − − j W (x) = exp( 2a2) exp( 3b2) exp( c2) exp( c2) exp( c2). M~ − − − 1 − 2 − 3

4.5 Absolute value of determinants

Though our Boltzmann factor integrand ΩM~ (x) is recognizably determinantal, we still need to remove the absolute value before we can apply Lemma 4.3 (and later Lemma 4.2). This can be done by decomposing the domain of integration into subsets over which the sign of the determinant is constant. Namely, we use totally ordered subsets ∆N (σ) over which the diﬀerences in the conﬂuent Vandermonde determinant never change signs. These smaller domains of integration are exactly the ones which allow us to apply (Chen’s) Lemma 4.2.

r As in section 3, suppose Lj is even for 1 j r. Let Ke = j=1 Mj be the total number of particles J ≤ ≤ with even charge, and let Ko = j=r+1 Mj be the total numberP of particles with odd charge. Relabel y = x1, y = x1, P y = x1 , y = x2 y = xr 1 1 2 2 ··· M1 M1 M1+1 1 ··· Ke Mr so that ~y = (x1, x2,..., xr) gives the locations of the particles of even charge. Similarly, relabel

r+1 r+1 r+2 J w1 = x , wM = x , wM +1 = x wK = x 1 ··· r+1 Mr+1 r+1 1 ··· e MJ r+1 r+2 J e so that ~w = (x , x ,..., x ) gives the locations of the particles of odd charge. Deﬁne λj to be the Lk which corresponds to yj so that

e e e Λ = (λ1,...,λKe ) = (L1,...,L1,L2,...,L2,...,Lr,...,Lr)

o gives the list of even charges, with each Lj appearing Mj times. Similarly deﬁne λj for corresponding wj so that o o o Λ = (λ1,...,λKo ) = (Lr+1,...,Lr+1,Lr+2,...,Lr+2,...,LJ ,...,LJ ) e o gives the list of the odd charges. We will treat all charges λj and λj as distinct until it is relevant to recall which charges are repeated (and how many times each).

For each (σ, τ) S S , deﬁne V L,~ M~ (x) to be the matrix ∈ Ke × Ko στ ~ ~ e e o o L,M λ −1 λ −1 λ −1 λ −1 V (x)= V σ (1) (y −1 ) V σ (Ke) (y −1 ) V τ (1) (w −1 ) V τ (Ko) (w −1 ) , σ,τ σ (1) ··· σ (Ke) τ (1) ··· τ (Ko) h i L~ ,M~ obtained from V (x) by permuting the columns so that the columns with yσ−1(1) come ﬁrst (of which e − e there are λσ−1 (1) many), then all of the columns with yσ 1(2) come next (of which there are λσ−1(2) many), and so on until the yj are exhausted, doing the same for the wj .

17 Using the same conﬂuent Vandermonde determinant identity from subsection 4.4, we get

Ke Ko e e o e L,~ M~ λ −1 λ −1 λ −1 λ −1 det V (x)= (y −1 y −1 ) σ (k) σ (j) (w −1 y −1 ) σ (k) τ (j) σ,τ σ (k) − σ (j) × σ (k) − τ (j) j

Next, consider x = (~y, ~w) ∆Ke (σ) ∆Ko (τ) in which the even charged particles (located by ~y) are ordered according to σ and the odd∈ charged× particles (located by ~w) are ordered according to τ. In particular, wτ −1(k) > wτ −1(j) whenever j < k. Thus, all differences in the third product are positive. Additionally, each e difference in the first and second products have even exponents λj . Thus,

L,~ M~ L,~ M~ det Vσ,τ (x) = det Vσ,τ (x)

on the domain ∆K (σ) ∆K (τ). e × o Note, permuting the variables ~y involves permuting blocks of even numbers of columns at a time, leaving the determinant of V L,~ M~ (x) unchanged. In contrast, permuting variables ~w involves permuting blocks of odd numbers of columns at a time, changing the determinant by sgn(τ). Thus,

L,~ M~ L,~ M~ L,~ M~ det V (x) = sgn(τ) det Vσ,τ (x) = det Vσ,τ (x)

L,~ M~ x L,~ M~ x on the domain ∆Ke (σ) ∆ Ko (τ). Analogous results hold if we replace V ( ) with H ( ) (as deﬁned in subsection 4.4). ×

Recall the example from subsection 4.4 in which we have a single particle of even charge 2. Then Ke = 1, ~y = (a), and Λe = (2). There is one particle of odd charge 3, and there are three particles of odd charge 1. o Then Ko = 4, ~w = (b,c1,c2,c3), and Λ = (3, 1, 1, 1). Let τ be the permutation which swaps b with c1. The new matrix (with permuted columns) is given by 10110 0 11 a 1 c1 b 1 0 c2 c3 a2 2a c2 b2 2b 1 c2 c2 L,~ M~ 1 2 3 3 2 3 3 2 3 3 Vid,τ (x)= a 3a c b 3b 3b c c   1 2 3  . .   . .    a7 7a6 c7 b7 7b6 21b5 c7 c7  1 2 3 Note, swapping two variables requires more than just swapping two columns. We swap the entire three- 3 1 column block V (b) with the one-column block V (c1).

5 Canonical ensemble

With the modiﬁcation to the integrand outlined in subsection 4.5, we can now decompose ZM~ into integrals without absolute value, provided we divide the domain of integration appropriately. Explicitly, 1 ZM~ = ΩM~ (x) dy1 dyKe dw1 dwKo M !M ! M ! RKe RKo ··· ··· 1 2 ··· J Z Z 1 L,~ M~ = det V (x) WM~ (x) dy1 dyKe dw1 dwKo M1!M2! MJ ! ··· ··· ∈ ∈ ∆Ke (σ) ∆Ko (τ) ··· σ SKe τ SKo Z Z X X 1 L,~ M~ = det Hσ,τ (x) dy1 dyKe dw1 dwKo , M1!M2! MJ ! ··· ··· ∈ ∈ ∆Ke (σ) ∆Ko (τ) ··· σXSKe τ XSKo Z Z

18 summing over all totally ordered subsets ∆ (σ) RKe and ∆ (τ) RKo . Ke ⊂ Ko ⊂

5.1 When all Lj are even

For π = (π ... π ) S and complete N-family of monic polynomials ~p = p N , deﬁne 1| | Ke ∈ N { n}n=1 e λk Qk (x)= exp( U(x)) Dl−1p (x), πj − · πj (l) Yl=1 so that we can write −1 −1 L,~ M~ σ (1) σ (Ke) det H (~y)= sgn(π)Q (yσ−1(1)) Q (yσ−1(K )). σ π1 ··· πKe e ∈ πXSN k e Note, each Qπj is a product of λk many matrix entries which we have grouped together. These entries are e taken from the λk many derivative columns for the one variable yk. Writing the determinant this way allows to invoke Lemma 4.3 which gives us

−1 −1 ¡ ~ ~ α ¡ α det HL,M (~y)= σ (1) ∧ ···∧ σ (Ke) dε , σ K ! vol Z e where each αk is deﬁned by

k k e αk(x)= At (x) εt = sgn(τ)Qt◦τ (x) εt = exp( λkU(x))Wr(~pt, x) εt. e e e − t:λ րN t:λ րN τ∈Sλe t:λ րN Xk Xk Xk Xk

As mentioned in subsection 4.4, these Wronskians are the determinants of the univariate Lj Lj minors of the conﬂuent Vandermonde matrix. Recall from section 3, ×

γj = Wr(~pt, x) dµj (x) εt, R t ր :LXj N Z with dµ (x) = exp( L U(x)) dx. Starting from j − j 1 L,~ M~ ZM~ = det Hσ (~y) dy1 dyKe , M1!M2! MJ ! ··· ∈ ∆Ke (σ) ··· σXSKe Z relabeling the variables xj = yσ−1(j) produces

1 L,~ M~ ZM~ = det Hσ (~x) dx1 dxKe . M1!M2! MJ ! ··· ∈ ∆Ke (id) ··· σXSKe Z By Lemma 4.3, the determinant can be rewritten in the exterior shuﬄe algebra. Thus,

−1 −1 ¡ 1 α ¡ α Z = σ (1) ∧ ···∧ σ (Ke) dε . M~ M !M ! M ! K ! vol 1 2 J ∈ e ··· σXSKe Z Applying (Chen’s) Lemma 4.2 sends integration of shuﬄe products to ordinary products of integrals. Thus, 1 γ∧M1 γ∧MJ Z = 1 ∧···∧ J dε . M~ M !M ! M ! K ! vol 1 2 J ∈ e ··· σXSKe Z e Note, there exist Mj many k for which λk = Lj, so each factor γj appears Mj times. This happens independent of σ S , of which there are S = K ! many. Thus, ∈ Ke | Ke | e 1 Z = γ∧M1 γ∧MJ dε . M~ M !M ! M ! 1 ∧···∧ J vol 1 2 ··· J Z We have now proven the following lemma:

19 Lemma 5.1. If all Lj are even, then

γ∧M1 γ∧MJ Z = 1 J dε . M~ M ! ∧···∧ M ! vol Z 1 J

5.2 Even number of odd charges

J In this subsection, we assume all Lj are odd, but the total number of particles Ko = j=1 Mj =2K is even. This happens, for example, when total charge N is even. Recall from section 3, P

η = Wr(~pt, x)Wr(~ps,y) dµ (x)dµ (y) εt εs. j,k j k ∧ t ր s ր x

j,k j,k αj,k = At εt = sgn(π)Qt◦π εt, o o o o t:λ +λ րN t:λ +λ րN π∈Sλo+λo j Xk j Xk Xj k where o o λj λk j,k l−1 l−1 Qt (x, y)= exp( U(x)) D pt◦π(l)(x) exp( U(y)) D pt◦π(λo+l)(y), ◦π − · − · j Yl=1 Yl=1 o o j,k then αj,k is an antisymmetrized (λj +λk)-form by construction. As before, we should think of Qt as taking o one entry from each of the λj many derivative columns for one variable x and then one entry from each of o the λk many derivative columns for the next variable y. Pairing an odd number of entries with another odd j,k number of entries produces an even form. It remains to be shown that the antisymmetrizations At are composed of complementary Wronskian minors.

o o L,~ M~ o o o o L,~ M~ For t : λ + λ N, let H t (w , w ) denote the (λ + λ ) (λ + λ ) minor of H ( ~w) comprised j k ր τ, j k j k × j k τ t t o o o o of rows (1),..., (λj + λk) taken from the λj columns in wj and the λk columns in wk. When viewed as a j,k two variable function, At is the determinant of this minor. Explicitly,

j,k L,~ M~ At (wj , wk) = det Hτ,t (wj , wk).

o o o L,~ M~ o o L,~ M~ For t1 : λ λ + λ , let H t (wj ) denote the λ λ minor of H t (wj , wk) comprised of rows j ր j k τ, 1 j × j τ, o o o o o L,~ M~ t1(1),..., t1(λ ) taken from the λ columns in wj . Similarly, for t2 : λ λ + λ , let H t (wk) denote a j j k ր j k τ, 2 λo λo minor in w . By the Laplace expansion of the determinant, k × k k L,~ M~ L,~ M~ L,~ M~ det H t (wj , wk) εt = det H t (wj ) det H t (wk) sgn(t1, t2) εt τ, τ, 1 τ, 2 · t :λoրλo+λo t :λo րλo+λo 1 j Xj k 2 kXj k L,~ M~ L,~ M~ = det H t (w ) det H t (w ) εt εt τ, 1 j τ, 2 k · 1 ∧ 2 t :λoրλo+λo t :λo րλo+λo 1 j Xj k 2 kXj k o o = exp( λ U(w ))Wr(~pt , w ) exp( λ U(w ))Wr(~pt , w ) εt εt . − j j 1 j − k k 2 k · 1 ∧ 2 t :λoրλo+λo t :λo րλo+λo 1 j Xj k 2 kXj k Thus, η is the result of applying to the two-variable coeﬃcients of α as desired. j,k h·i2 j,k

20 Proceeding as we did in subsection 5.1 (applying Lemma 4.3 in the third line below and Lemma 4.2 in the last line),

1 L,~ M~ ZM~ = det Hτ ( ~w) dw1 dwKo M1!M2! MJ ! ··· ∈ ∆Ko (τ) ··· τ XSKo Z 1 L,~ M~ = det Hτ (~x) dx1 dxKe M1!M2! MJ ! ··· ∈ ∆Ko (id) ··· τ XSKo Z

−1 −1 −1 −1 ¡ 1 α ¡ α − = τ (1),τ (2) ∧ ···∧ τ (Ko 1),τ (Ko) dε M !M ! M ! K! vol 1 2 J ∈ ··· τ XSKo Z J J 1 1 ∧ j,k = η Mτ dε , M !M ! M ! K! j,k vol 1 2 J ∈ j=1 ··· τ XSKo Z ^ k^=1 j,k o o j,k where Mτ is the number of times λτ −1(2n−1) = Lj while λτ −1(2n) = Lk, and K = Ko/2= j,k Mτ is the total number of factors in the wedge product. Note, these M j,k exponents depend on τ, so we’re not able to τ P drop the sum. However, the Mj many particles with the same charge Lj are indistinguishable. Restricting to shuﬄe permutations removes the redundancy in permuting variables which have the same Lj. We have now proven another lemma:

Lemma 5.2. If all Lj are odd, but total charge N is even, then

J J 1 ∧ j,k Z = η Mτ dε . M~ K! j,k vol j=1 τ∈Sh(MX1,...,MJ ) Z ^ k^=1

Recall the example from subsection 4.4 and subsection 4.5. Modify this example by replacing the even charge 2 particle with an odd charge 3 particle, leaving the other particle of odd charge 3 and three particles of odd charge 1. The three columns in variables a and b produce 3 3 Wronskian minors, while the remaining columns in the variables c ,c ,c produce 1 1 Wronskian minors.× 1 2 3 × Under the identity permutation, we pair the three columns in variable a with the three columns in variable b to produce η3,3. Pairing the one column in variable c1 with the one column in variable c2 produces η1,1.

Under the permutation τ which previously swapped the three columns in variable b with the one column in variable c1, we pair variable a (charge 3) with c1 (charge 1), and we pair b (charge 3) with c2 (charge 1). After integrating out all the variables, the result is two copies of η3,1.

The permutation which swaps a with b produces the same η3,3 as the identity permutation. To avoid this redundancy, we consider only shuﬄe permutations. The permutation which moves c1 to the front (ordering the variables as c1,a,b,c2,c3) produces η1,3 followed by the distinct η3,1.

Note, in this example as stated, the last variable c3 is unpaired because we have an odd number of variables. In the next subsection, we demonstrate how to ﬁx this. Amending an extra column to the conﬂuent Vandermonde matrix allows us to pair the last variable with a placeholder. Once integrated, this last “pair” produces the single Wronskian form γj instead of the double Wronskian form ηj,k.

21 5.3 Odd number of odd charges

In this subsection, we again assume all L are odd, but the total number of particles K = J M =2K 1 j o j=1 j − L,~ M~ is odd. This happens, for example, when total charge N is odd. Start by modifying Hτ P( ~w) as follows: ~ ~ ~ ~ HL,M ( ~w) 0 HL,M,1( ~w)= τ τ 0 1 In the previous subsection, we constructed even forms αj,k and subsequent ηj,k by pairing an Lj charge with an adjacent Lk charge. We also showed taking determinants of appropriate minors produces an antisymmetrized form αj,k (for which our Lemma 4.3 will apply). We will do this pairing again for the ﬁrst 2K 2 j,k j,k − variables, which makes K 1 pairs. Explicitly, simply deﬁne Qt , At , α , and η as before. − j,k j,k

o − For convenience, let L = λ −1 be the charge corresponding to the last variable w 1 . Construct τ (Ko) τ (Ko) L,~ M,~ 1 −1 the ﬁnal ατ (Ko) by taking (L+1) (L+1) minors from the last L+1 columns of Hτ ( ~w). These minors have non-zero determinant only when× the last row is chosen. Thus, valid minors are entirely determined by a choice of only L many other rows, and

−1 τ (Ko) α −1 = sgn(π)Qt εt ε τ (Ko) ◦π ∧ N+1 t ր ∈ :LXN πXSL is an antisymmetrized (L + 1)-form when

L −1 τ (Ko) l−1 Qt (x)= exp( U(x)) D p (x). ◦π − · π(l) Yl=1

−1 −1 As in subsection 5.1, applying 1 to ατ (Ko) produces γτ (Ko) εN+1. Then the expression for ZM~ changes only slightly to includeh·i this new (unpaired, single Wronskian)∧ factor:

Lemma 5.3. If all Lj are odd, and the total charge N is odd, then

J J j,k 1 ∧Mτ Z = η γ −1 dε , M~ K! j,k ∧ τ (Ko) vol j=1 τ∈Sh(MX1,...,MJ ) Z ^ k^=1 j,k where K = (Ko + 1)/2=1+ j,k Mτ is the total number of factors in the wedge product. P Recall the example at the end of the previous subsection with two particles of charge 3 and three particles of charge 1. Under the identity permutation, we pair a charge 3 with a charge 3, pair a charge 1 with a charge 1, and leave a charge 1 unpaired. This produces η3,3 η1,1 γ1. Under the permutation τ which swapped the second charge 3 with the ﬁrst charge 1, we got η ∧ η ∧ γ . 3,1 ∧ 3,1 ∧ 1 Consider instead the permutation which puts all of the charge 1 particles before the charge 3 particles. We pair a charge 1 with a charge 1, pair the last charge 1 with a charge 3, and leave a charge 3 unpaired. This produces η η γ . 1,1 ∧ 1,3 ∧ 3

5.4 Arbitrary charge vector

Finally, we allow any mix of odd and even charges. Recall (from the beginning of section 5),

1 L,~ M~ ZM~ = det Hσ,τ (x) dy1 dyKe dw1 dwKo . M1!M2! MJ ! ··· ··· ∈ ∈ ∆Ke (σ) ∆Ko (τ) ··· σXSKe τ XSKo Z Z

22 r J Let Ne = j=1 LjMj be the total charge of the even charges, and let No = j=r+1 Lj Mj be the total charge of the odd charges. By the Laplace expansion of the determinant, P P L,~ M~ L,~ M~ L,~ M~ det H (~y, ~w) ε = det H t (~y) det H s ( ~w) εt εs, σ,τ vol σ, τ, · ∧ t ր s ր :NXe N :NXo N

L,~ M~ where det H t (~y) is an N N minor taken only from columns in the (even charge) variables ~y, and σ, e × e L,~ M~ det Hσ,s ( ~w) is an No No minor taken only from columns in the (odd charge) variables ~w. Note, εt εs =0 whenever these minors× are not complimentary. ∧

With the variables separated in this way, we can apply Lemma 5.1 to the determinant in the even charges, and we can apply either Lemma 5.2 or Lemma 5.3 to the determinant in the odd charges. For the even charges, we have

t ∧M1 t∧Mr 1 L,~ M~ γ1 γr det Hσ,t (~y) dy1 dyKe εt = , M1! Mr! ··· · M1! ∧···∧ Mr! ∈ ∆Ke (σ) ··· σXSKe Z t L,~ M~ where γ is subtly diﬀerent from γ because H t (~y) is already an N N minor chosen by t. Explicitly, j j σ, e × e

t γj = Wr(~pt◦tj , x) dµj (x) εtj . R t ր j :LXj Ne Z Taking the sum over all t : N N gives us back the original γ forms e ր j

t ∧M t∧Mr γ 1 γ γ∧M1 γ∧Mr 1 r = 1 r . M ! ∧···∧ M ! M ! ∧···∧ M ! t ր 1 r 1 r :NXe N It is straightforward to check an analogous result holds for the determinant in the odd charges. The following lemma supersedes Lemmas 5.1, 5.2, and 5.3: Lemma 5.4. Suppose L is even for 1 j r, then when N is even, j ≤ ≤

∧M1 ∧M J J γ γ r 1 ∧ j,k Z = 1 r η Mτ dε , M~ M ! ∧···∧ M ! ∧ K! j,k vol 1 r j=1 Z τ∈Sh(MXr+1,...,MJ ) ^ k^=1 and when N is odd,

J J ∧M1 ∧Mr j,k γ1 γr 1 ∧Mτ Z = η γ −1 dε . M~ M ! ∧···∧ M ! ∧ K! j,k ∧ τ (Ko) vol 1 r j=1 Z τ∈Sh(MXr+1,...,MJ ) ^ k^=1

6 Isocharge grand canonical ensemble

Recall from subsection 1.3, we want to compute

Z = zM1 zM2 zMJ Z . N 1 2 ··· J M~ L~ ·XM~ =N

23 6.1 When all Lj are even

Starting from Lemma 5.1, we have

∧M1 ∧MJ M1 MJ γ1 γJ ZN = z1 zJ dεvol. ··· M1! ∧···∧ MJ ! L~ ·XM~ =N Z Recall from subsection 2.2, the Berezin integral is a projection onto the highest exterior power N (RN ). If each γ is an L -form, then the wedge product above is an L~ M~ -form. If we extend the sum over all M~ , the j j · V Berezin integral will eliminate any summands for which L~ M~ = N. Thus, · 6 ∞ ∞ ∧M1 ∧MJ (z1γ1) (zJ γJ ) ZN = dεvol ··· M1! ∧···∧ MJ ! Z MX1=0 MXJ =0 ∞ J (z γ )∧M = j j dε M! vol j=1 Z ^ MX=1 = exp(z γ ) exp(z γ ) dε 1 1 ∧···∧ J J vol Z = BE (z γ + + z γ ). vol 1 1 ··· J J In the last line, we replace the product of these exponentials with the exponential of the sum, which we can do because our forms are even and therefore commute. This completes the proof of Theorem 3.1.

6.2 When all Lj are odd

Let’s start by assuming there are no even species, and the total charge N is even. Recall Lemma 5.2 which gives us J J 1 ∧M j,k Z = zM1 zMJ η τ dε . N 1 ··· J K! j,k vol j=1 L~ ·XM~ =N τ∈Sh(MX1,...,MJ ) Z ^ k^=1

For M~ ﬁxed and τ Sh(M1,...,MJ ), the number of other permutations which produce the same pairs (j, k) is ∈ J J 1 K! j,k . j=1 Mτ ! Y kY=1 j,k This is just a multinomial coeﬃcient, recalling K is the sum of the Mτ . Of these permutations, there exists a unique representative which orders the pairs (j, k) lexicographically. Let M~ be the set of these representatives, then L

∞ ∞ J J ∧ j,k (z z η ) Mτ Z = j k j,k dε . N ··· j,k vol M =0 M =0 τ∈L j=1 k=1 Mτ ! Z X1 XJ XM~ ^ ^ Next, we condition on population vectors M~ which produce the same K, the number of (j, k) pairs, and so

∞ J J ∧M j,k (zj zkηj,k) τ ZN = j,k dεvol. K=0 M +···+M =2K τ∈L j=1 k=1 Mτ ! Z X 1 X J XM~ ^ ^

24 Collecting these together, we get the Kth power of the sum over all possible pairs (j, k). Thus,

∧K ∞ 1 J J Z = z z η dε N K!  j k j,k vol j=1 Z KX=0 X Xk=1 J J  = BE z z η . vol  j k j,k j=1 X kX=1   In the case with total charge N odd, we can go through the same steps starting from Lemma 5.3. This produces J J J Z = BE z z η + z γ ε . N vol1  j k j,k j j ∧ N+1 j=1 j=1 X kX=1 X   Note, when N is odd, every ZM~ has exactly one γj for each τ (see Lemma 5.3). The εN+1 shown attached to each of the γj above ensures this is the case when we exponentiate and take the Berezin integral. First, ε ε = 0, so γ ε γ ε = 0. Thus, terms in the expansion of the exponential with N+1 ∧ N+1 j ∧ N+1 ∧ k ∧ N+1 more than one γj are annihilated. Because the Berezin integral with respect to dεvol1 projects onto the N+1 N+1 highest exterior power (R ), terms in the expansion of the exponential with no γj are missing the basis vector εN+1 and are annihilated by the Berezin integral (with respect to dεvol ). Thus, we only get V 1 summands (in the expansion of the exponential) with exactly one γj , as in Lemma 5.3.

6.3 Full generalization

Recall Lemma 5.4, in which the γ1,...,γr corresponding to the even charges are already factored out. Summing over all possible M ,...,M , we can factor out an exp(z γ + z γ ) as in subsection 6.1. From 1 r 1 1 ··· r r what remains, we obtain the exponential of the sum of the ηj,k, possibly with an extra set of γj εN+1 forms. For N even, ∧

∧K r ∞ ∞ J J (z γ )∧M 1 Z = j j z z η dε N  M!  ∧ K!  j k j,k vol j=1 j=1 Z ^ MX=1 KX=0 X Xk=1  r  J J  = exp z γ exp z z η ε  j j  ∧  j k j,k vol j=1 j=r+1 Z X X k=Xr+1  r  J J  = BE z γ + z z η . vol  j j j k j,k j=1 j=r+1 X X k=Xr+1   This concludes the proof of Theorem 3.2 and, with a slight modiﬁcation, Theorem 3.3.

7 Circular ensembles

Returning to the original setup in subsection 1.3, consider instead charged particles on the unit circle. Substitute all instances of R with [0, 2π) so that

x = (x1, x2,..., xJ ) [0, 2π)M1 [0, 2π)M2 [0, 2π)MJ ∈ × ×···×

25 j gives the locations of the particles around the unit circle with each xn [0, 2π) corresponding to an angle. Assuming logarithmic interaction between the particles, the energy contributed∈ by interaction between two k j particles of charge L and L at angles xj and xk respectively is given by L L log eixn eixm . Thus, j k n m − j k − at inverse temperature β, the total potential energy of the system is given by

J Mj Mk j j k j E (x)= β L2 log eixn eixm β L L log eixn eixm , M~ − j − − j k − j=1 m

J 2 Mj Mk j j βLj k j βLj Lk Ω (x) = exp( E (x)) = eixn eixm eixn eixm . M~ − M~ − × − j=1 m

L~ ,M~ Ω (x) det V (y) ρ (x)= M~ = , M~ ZM~ M1!M2! MJ ! ZM~ M1!M2! M J ! ··· ··· with partition function

1 L,~ M~ M1 1 M2 2 MJ J ZM~ = det V (y) dν (x ) dν (x ) dν (x ), M1!M2! MJ ! [0,2π)M1 ··· [0,2π)MJ ··· ··· Z Z where νMj is Lebesque measure on [0, 2π)Mj . Using the same modifications as before in the linear case, we can assume β = 1 for computational purposes. Note, the same confluent Vandermonde determinant gives us the same product of differences (with exponents) as before, even with the new complex variables y in place of the real variables x.

7.1 Complex modulus

We will be able to apply our same algebraic lemmas to our determinantal integrand once we resolve the absolute value. As observed in [14], the absolute diﬀerence can be decomposed as

k j −i xk +xj /2 k j eixn eixm = ie ( n m) eixn eixm sgn(xk xj ) − − − n − m k j −i xk +xj /2 ixk ixj xn xm = ie ( n m) e n e m − . − − k j x xm n −

Next, we deﬁne L T/2 dµ (x)= ie−ix j dx, j − where J T = L + L M , − j k k Xk=1 − j ~ ~ so that we can bring the (complex valued) weight functions ( ie ixm )T/2 inside the matrix V L,M (y) the − same way we did in subsection 4.4. Explicitly, construct HL,~ H~ (y) by multiplying each column with the − j variable xj by ( ie ixm )T/2. Note, there will be L many columns for each xj . Finally, we can write the m − j m (xk xj )Lj Lk / xk xj Lj Lk factors as separate conﬂuent Vandermonde determinants in x. n − m | n − m|

26 We can use the same procedure of separating the odd species from the even species and decomposing the integral over ordered subsets as in subsection 4.5. Thus,

1 L,~ M~ ZM~ = det V (y) dy1 dyKe dw1 dwKo M1!M2! MJ ! [0,2π)Ke [0,2π)Ko ··· ··· ··· Z Z L,~ M~ 1 L,~ M~ det V (x) = det H (y) dy1 dyKe dw1 dwKo . M1!M2! MJ ! L,~ M~ ··· ··· ∈ ∈ ∆Ke (σ) ∆Ko (τ) det V (x) ··· σXSKe τ XSKo Z Z

Observe ~ ~ det V L~ ,M~ (x) sgn(τ) det V L,M (x) L~ ,M~ y L,~ M~ y σ,τ 2 L,~ M~ y det H ( ) = sgn(τ) det Hσ,τ ( ) ~ ~ = sgn(τ) det Hσ,τ ( ) L,~ M~ L,M det V (x) sgn(τ) det Vσ,τ (x) for y ∆ (σ) ∆ (τ). Thus, ∈ Ke × Ko

1 L,~ M~ ZM~ = det Hσ,τ (y) dy1 dyKe dw1 dwKo . M1!M2! MJ ! ··· ··· ∈ ∈ ∆Ke (σ) ∆Ko (τ) ··· σXSKe τ XSKo Z Z

7.2 Statement of results

Proceeding through the same methods presented in section 5 and section 6, we get the same theorems from section 3 with slight modification to our forms γj and ηj,k. Given a complete N-family of monic polynomials, define 2π ix γj = Wr(~pt,e ) dµj (x) εt, t ր 0 :LXj N Z and define

ix iy η = Wr(~pt,e )Wr(~ps,e ) dµ (x)dµ (y) εt εs, j,k j k ∧ t ր s ր 0

A Appendix

Let Λ=(λ1,...,λK ) be a partition of N. Recall from subsection 4.4 the definitions of the Young subgroup H(Λ) SN , the subset of block permutations Bl(Λ) SN , the subset of shuffle permutations Sh(Λ), and the subset⊆ of ordered shuffle permutations Sh◦(Λ). ⊂

Here we provide a proof of Lemma 4.1.

Lemma 4.1. Let Λ = (λ1,...,λK ) be a partition of N. Given any ϕ SN , there exists unique permutations τ H(Λ), π Bl(Λ), and σ Sh◦(Λπ) so that ϕ = σ π τ. ∈ ∈ ∈ ∈ ◦ ◦

Conducive to this proof, it will be convenient to give alternate definitions for the different subsets of j−1 permutations. First, define sj = k=1 λk to be the partial sums of the λk, up to but not including λj so P

27 that s1 = 0, s2 = λ1, s3 = λ1 + λ2, and so on. We alternatively deﬁne the Young subgroup H(Λ) to be the σ SN such that ∈ s +1 σ(s + j) s k ≤ k ≤ k+1 for all 1 k K and 1 j λ . We deﬁne the block permutations Bl(Λ) to be the σ S such that ≤ ≤ ≤ ≤ k ∈ N

σ(sk + j)+1= σ(sk + j + 1)

for all 1 k K and 1 j λk. Recall the deﬁnition of θσ SK from subsection 4.4. This is the unique permutation≤ ≤ such that ≤ ≤ ∈

σ(s −1 + 1) < σ(s −1 + 1) < < σ(s −1 + 1). θσ (1) θσ (2) ··· θσ (K) The shuﬄe permutations Sh(Λ) are the σ S such that ∈ N

σ(sk + i) < σ(sk + j)

for all 1 k K and 1 i < j λ . The ordered shuﬄe permutations Sh◦(Λ) additionally satisfy ≤ ≤ ≤ ≤ k

σ(sj + 1) < σ(sk + 1)

for all 1 j < k K. ≤ ≤ Demonstrably, H(Λ) = K S = λ ! λ !, and Bl(Λ) = S = K!. Heuristically, a shuffle | | k=1 | λk | 1 ··· K | | | K | permutation σ Sh(Λ) is constructed by choosing from N positions the location of the first λ1 elements, ∈ Q then the next λ2 elements, and so on until all elements are exhausted. It is straightforward to see Sh(Λ) is the multinomial coefficient | | N N! Sh(Λ) = = . | | λ ,...,λ λ ! λ ! 1 K 1 ··· K We prove Lemma 4.1 in two steps. First, we show the decomposition of an arbitrary permutation into a product of a shuffle permutation after a permutation belonging to the Young subgroup. Second, we show this shuffle permutation can be further decomposed into a product of an ordered shuffle permutation after a block permutation. Lemma A.1. Given any ϕ S , there exists unique τ H(Λ) and ρ Sh(Λ) so that ϕ = ρ τ. ∈ N ∈ ∈ ◦

Proof. Consider the collection of right cosets S /H(Λ). For each 1 k K and 1 j λ , let ak be the N ≤ ≤ ≤ ≤ k j jth smallest element of the set ϕ(s + 1), ϕ(s + 2),...,ϕ(s + λ ) , and deﬁne a permutation τ S by { k k k k } ∈ N −1 k τ(sk + j)= ϕ (aj )

for 1 k K and 1 j λ . Then τ H(Λ), and ≤ ≤ ≤ ≤ k ∈ ϕ τ(s + i) < ϕ τ(s + j) ◦ k ◦ k whenever 1 i < j λ . Thus, every coset T S /H(Λ) contains at least one shuffle permutation. Note, ≤ ≤ k ∈ N N! S /H(Λ) = = Sh(Λ) . | N | λ ! λ ! | | 1 ··· K Thus, each coset contains a unique shuffle permutation. Define ρ Sh(Λ) to be this unique shuffle permutation in the coset to which ϕ belongs. Then ϕ = ρ τ as desired.∈ ◦ Lemma A.2. Given any ρ Sh(Λ), there exists unique permutations π Bl(Λ) and σ Sh◦(Λπ) so that ρ = σ π. ∈ ∈ ∈ ◦

28 Proof. For each 1 k K, deﬁne b to be the kth smallest element of the set ≤ ≤ k ρ(s + 1),ρ(s + 1),...,ρ(s + 1) . { 1 2 K } Let α S be the permutation satisfying ∈ K

ρ(sk +1)= bα(k) for 1 k K. Deﬁne π Bl(Λ) by ≤ ≤ ∈ π(s + j)= ρ−1(b )+ j 1 k α(k) − for 1 k K and 1 j λ . Then α = θ . Let σ = ρ π−1. We want to show σ Sh◦(Λπ). ≤ ≤ ≤ ≤ k π ◦ ∈

Let µ =Λπ, and let t = j−1 µ . Suppose 1 k K and 1 j µ . Observe that j k=1 k ≤ ≤ ≤ ≤ k P σ(t + j)= ρ π−1(t +1+ j 1) k ◦ k − −1 = ρ π (π(s −1 +1)+ j 1) ◦ α (k) − −1 = ρ π (π(s −1 + j)) ◦ α (k) = ρ(sα−1(k) + j).

Thus, σ(tk + j)= ρ(sα−1(k) + j).

If 1 i < j µk, then ρ(sα−1(k) + i) < ρ(sα−1(k) + j) since ρ Sh(Λ). Thus, σ(tk + i) < σ(tk + j), and σ Sh(Λ≤ π). ≤ ∈ ∈ Next, if i < j, then ρ(sα−1(i) +1)= bi

ρ(sk +1)= bα(k)

for 1 k K. Deﬁne c to be the kth smallest element of the set π′(s + 1), π′(s + 1),...,π′(s + 1) . ′ k 1 2 K ≤′ ≤ ◦ π ′ { } Since σ Sh (Λ ), we have σ (ck)= bk for each 1 k K, and thus θπ′ = α = θπ. Since π is completely determined∈ by θ , we have π = π′. Thus, σ′ = π−1 ≤ρ =≤σ as desired. π ◦

Finally, Lemma 4.1 follows immediately from applying Lemma A.2 after Lemma A.1.

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