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Elliptic hypergeometric integrals Eric M. Rains Department of Mathematics California Institute of Technology MPIM Oberseminar, Bonn, 24/7/2008

Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

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Page 1: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Elliptic hypergeometric

integrals

Eric M. Rains

Department of Mathematics

California Institute of Technology

MPIM Oberseminar, Bonn, 24/7/2008

Page 2: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Hypergeometric integrals

Gaussian integral:∫ ∞

−∞e−x2/2dx =

√2π

(Hermite polynomials)

Gamma integral:∫ ∞

0xα−1e−xdx = Γ(α)

(Laguerre polynomials)

Beta integral:

∫ 1

0xα−1(1 − x)β−1dx =

Γ(α)Γ(β)

Γ(α + β)

(Jacobi polynomials)

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Page 3: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Why “hypergeometric”?

∫ 1

0xα−1(1 − x)β−1(1 − tx)−γdx

=∑

0≤k

Γ(β)Γ(α + k)Γ(γ + k)

Γ(γ)Γ(α + β + k)Γ(1 + k)tk

=Γ(α)Γ(β)

Γ(α + β)2F1(α, γ;α + β; t)

Note transformations:

2F1(a, b; c; t) = 2F1(b, a; c; t)

= (1 − t)c−a−b2F1(c − b, c − a; c; t)

= (1 − t)−b2F1(c − a, b; c;

−t

1 − t)

and evaluation

2F1(a, b; c; 1) =Γ(c)Γ(c − a − b)

Γ(c − a)Γ(c − b)

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Page 4: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Multivariate analogues

(Selberg) Let α, β, τ be complex numbers with

positive real parts. Then

1

n!

[0,1]n

1≤i<j≤n

|xi − xj|2τ∏

1≤i≤n

xα−1i (1 − xi)

β−1dxi

=∏

0≤k<n

Γ((k + 1)τ)Γ(α + kτ)Γ(β + kτ)

Γ(τ)Γ(α + β + kτ)

(Heckman-Opdam Jacobi polynomials)

(Morris) Let ℜ(τ) > 0, ℜ(a + b) > −1. Then

1

n!

Tn

1≤i<j≤n

|zi − zj|2τ∏

1≤i≤n

z(a−b)/2i |1 + zi|a+b

=∏

0≤k<n

Γ((k + 1)τ)Γ(kτ + a + b + 1)

Γ(τ)Γ(kτ + a + 1)Γ(kτ + b + 1)

(a = b = 0: Jack polynomials)

These integrals appear frequently in random

matrix theory (esp. τ = 1/2,1,2)

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Page 5: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

(Dirichlet) Let α0, . . . , αn be complex numbers

with positive real parts. Then

xi=1

i

xαi−1i dxi =

i Γ(αi)

Γ(∑

αi)

When α are real, a probability distribution (“mul-

tivariate beta”)

(Anderson; earlier by Varchenko, much earlier

by Dixon): Let α0, . . . , αn be complex numbers

with positive real parts, and let a0 > · · · > an.

Then∫

xi∈[ai+1,ai]

1≤i<j≤n

|xi − xj|∏

1≤i≤n0≤j≤n

|aj − xi|αj−1dxi

=∏

0≤i<j≤n

(ai − aj)αi+αj−1

i Γ(αi)

Γ(∑

i αi)

4

Page 6: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Proof of Dirichlet integral: multiply by∫

u(∑

αi)−1 exp(−u)du = Γ(∑

αi)

and change variables xi = yi/u. Proof of Dixon-

Anderson integral reduces to Dirichlet integral

by another change of variables; Selberg inte-

gral follows (per Anderson) by integrating

0≤i<j≤n

(xi − xj)∏

1≤i<j≤n

(yi − yj)

0≤i≤n

xα−1i (1 − xi)

β−1∏

0≤i≤n1≤j≤n

|xi − yj|τ−1

over 0 ≤ xn ≤ yn ≤ · · · ≤ x1 ≤ y1 ≤ x0 ≤ 1.

Theory of orthogonal polynomials for Dirichlet

or Dixon-Anderson less satisfying, though. But

the latter is a useful tool nonetheless.

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Probabilistic interpretation of Dixon-Anderson

Integer αi: take a Hermitian matrix with char-

acteristic polynomial∏

i(λ−ai)αi and restrict to

a random hyperplane; Dixon-Anderson is dis-

tribution of new eigenvalues.

Similarly for a real symmetric matrix with char-

acteristic polynomial∏

i(λ − ai)2αi.

Due to Baryshnikov in complex case with αi ≡1. General complex case follows either by gen-

eralizing the argument or by taking the limit

as eigenvalues coalesce.

Note that the general case of Dixon-Anderson

follows by (nontrivial) analytic continuation from

the integer case. And the α ≡ 1 case is easy,

so this gives a (much more complicated) alter-

nate proof.

There are similar random matrix interpreta-

tions for exponential and Gaussian versions of

Dixon-Anderson.

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Page 8: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Increasing subsequences

For a permutation π ∈ Sn, an increasing sub-

sequence is a subset S ⊂ {1,2, . . . , n} with π

increasing on S. Let ℓ(π) be the maximum

size of an increasing subsequence of π.

Theorem (Gessel, Rains) The number of π ∈Sn with ℓ(π) ≤ k is

EU∈U(k)|Tr(U)|2n.

If we choose n randomly (Poisson: probability

e−λλ2n/n!), then choose π ∈ Sn uniformly at

random, the probability that ℓ(π) ≤ k is

EU∈U(k) exp(−2λℜ(Tr(U))).

This looks like the Morris integral (a=b=0,

τ = 1), with an extra factor∏

exp(−λ(zi +

1/zi)).

7

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Similar results apply to increasing (or decras-

ing) subsequences of fixed-point-free involu-

tions. Many other combinatorial models give

rise to Selberg integrals or discrete analogues

(first-passage percolation, polynuclear growth,

totally asymmetric exclusion, domino/lozenge

tilings, plane partitions). Typically τ = 1, al-

though symmetric models may have τ = 1/2,

τ = 2.

Many of these come from the Jack polynomial

identity

P(n)λ (1,1, . . . ,1; τ)

=∏

1≤i<j≤n

Γ(λi − λj + (j − i + 1)τ)Γ((j − i)τ)

Γ(λi − λj + (j − i)τ)Γ((j − i + 1)τ)

(for λ large, proportional to (λi − λj)τ)

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Page 10: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

q-analogues

Define

Γq(x) :=∏

0≤i

(1 − qix)−1 =: 1/(x; q)

Γq(x, y, . . . , z) := Γq(x)Γq(y) . . .Γq(z)

(Will use similar convention for other Γ func-

tions)

Note

Γq(qx) = (1 − x)Γq(x)

limq→1

(1 − q)−aΓq(qa)

(1 − q)−bΓq(qb)=

Γ(a)

Γ(b)

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Page 11: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Macdonald-Morris integral:

1

n!

Tn

1≤i<j≤n

Γq(t(zi/zj)±1)

Γq((zi/zj)±1)

1≤i≤n

Γq(azi, qb/zi)

Γq(zi, q/zi)

=∏

0≤i≤n−1

Γq(tiqab, ti+1, q)

Γq(tiqa, tiqb, t)

(a = b = 1: Macdonald polynomials)

On a circle of radius qk, the integrand con-

verges to 0 as k → ∞. Obtain expression as

a sum over residues: a discrete q-analogue of

the Selberg integral.

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Page 12: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Macdonald polynomials

The Macdonald polynomials P(n)λ (x1, . . . , xn; q, t)

are symmetric (invariant under permutationsof x1,. . . ,xn), have “leading” monomial

1≤i≤n xλii ,

and are orthogonal w.r.to Macdonald-Morrisfor a = b = 1.

Macdonald “conjectures”:

1 Explicit formula for P(n)λ (1, t, . . . , tn−1; q, t).

2 Explicit formula for inner product.

3 Symmetry:

P(n)λ (. . . , qµitn−i, . . . ; q, t)

P(n)λ (. . . , tn−i, . . . ; q, t)

is symmetric in λ, µ.

These generalize to arbitrary (finite) root sys-tems (Cherednik).

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Page 13: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

(Rahman)

1

2Γq(q)

z∈S1

0≤r≤4 Γq(trz±1)

Γq(Tz±1, z±2)

=

0≤r<s≤4 Γq(trts)∏

0≤r≤4 Γq(T/tr)

where T = t0t1t2t3t4. (t4 = 0: Askey-Wilson)

(Gustafson)

Γq(t)n

Γq(q)n2nn!

Tn

1≤i<j≤n

Γq(tz±1i z±1

j )

Γq(z±1i z±1

j )

1≤i≤n

0≤r≤4 Γq(trz±1i )

Γq(t2n−2Tz±1i , z±2

i )

=∏

0≤j<n

Γq(tj+1)∏

0≤r<s≤4 Γq(tjtrts)∏

0≤r≤4 Γq(t2n−2−jT/tr)

(t4 = 0: Koornwinder polynomials; these also

satsify Macdonald conjectures (Noumi, van Diejen,

Sahi))

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Page 14: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Gustafson’s proof

First, the identity:

1

Γq(q)n2nn!

z∈Tn

1≤i<j≤n

1

Γq(z±1i z±1

j )

1≤i≤n

0≤r≤2n+2 Γq(trz±1i )

Γq(Tz±1i , z±2

i )

=

0≤r<s≤2n+2 Γq(trts)∏

0≤r≤2n+2 Γq(T/tr)

where T =∏

0≤r≤2n+2 tr

Using this, evaluate an integral on Tn×Tn+1 in

two different ways, both of which give Selberg

analogue; then solve the recurrence.

Deja vu. . . This is an analogue of Dixon-Anderson

integral!

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Page 15: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Elliptic analogues

Ruijsenaars’ “elliptic Gamma function”:

Γp,q(x) =∏

0≤j,k

1 − pj+1qk+1/x

1 − pjqkx

Why elliptic? Consider

θp(x) :=Γp,q(qx)

Γp,q(x)=

0≤k

(1 − pkx)(1 − pk+1/x);

observe that

θp(x) = −xθp(px),

so θp is a theta function on the elliptic curve

C∗/p.

Also note

Γp,q(px) = θq(x)Γp,q(x)

Γp,q(x) = Γp,q(pq/x)−1

Γ0,q(x) = Γq(x).

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Page 16: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Spiridonov (“elliptic beta integral”):

1

2Γp(p)Γq(q)

z∈S1

0≤r≤5 Γp,q(trz±1)

Γp,q(z±2)

=∏

0≤r<s≤5

Γp,q(trts),

where∏

tr = pq.

Theorem (conjectured by van Diejen/Spiridonov;

“elliptic Selberg integral”):

Γp,q(t)n

Γp(p)nΓq(q)n2nn!

z∈Tn

1≤i<j≤n

Γp,q(tz±1i z±1

j )

Γp,q(z±1i z±1

j )

1≤i≤n

0≤r≤5 Γp,q(trz±1i )

Γp,q(z±2i )

=∏

0≤j<n

Γp,q(tj+1)

0≤r<s≤5

Γp,q(tjtrts),

where t2n−2 ∏

tr = pq.

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Page 17: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Also, following Gustafson (“elliptic Dixon-Anderson

integral”):

1

Γp(p)nΓq(q)n2nn!

z∈Tn

1≤i<j≤n

1

Γp,q(z±1i z±1

j )

1≤i≤n

0≤r≤2n+3 Γp,q(trz±1i )

Γp,q(z±2i )

=∏

0≤r<s≤2n+3

Γp,q(trts)

where∏

tr = pq.

Actually have transformation: m-dimensional

integral with 2m+2n+4 parameters transforms

to n-dimensional integral with related parame-

ters.

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Page 18: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

What about orthogonal polynomials? Already

a no-go theorem at the univariate level (Askey-

Wilson polynomials are the most general hy-

pergeometric orthogonal polynomials). But some-

thing slightly weaker works: biorthogonal el-

liptic functions. (Spiridonov/Zhedanov at the

elliptic level)

Can we make this work at the multivariate

level?

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Page 19: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

First key idea: double integral proof of Type II

integral should give adjoint integral operators.

Dixon-Anderson case (τ = 1):

Evsλ((1 − vv†)A(1 − vv†)) ∝ sλ(A)

so Dixon-Anderson integral takes n−1 variable

Schur functions to n variable Schur functions.

(Similar to Okounkov’s integral representation

for interpolation polynomials)

In general, elliptic Dixon-Anderson integral takes

n−1 variable biorthogonal functions to n-variable

biorthogonal functions. (Preservation of biorthog-

onality is easy; the hard part is showing that

the image is in the correct space: use a degen-

erate case of the transformation!)

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Page 20: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Second key idea: another proof replaces dou-

ble integral by difference operators

Alternate “raising” difference operator with “rais-

ing” integral operator; obtain a family of biorthog-

onal functions.

Two of three analogues of Macdonald’s con-

jectures are immediate! Proof of remaining

Macdonald conjecture (symmetry) uses extra

properties of “interpolation functions” (a spe-

cial case of the biorthogonal functions).

Taking suitable limits gives the Macdonald con-

jectures for Koornwinder and (ordinary) Mac-

donald polynomials.

Big open question: Other root systems?

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Page 21: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Type II transformations

Define

II(n)

(t0, . . . , t7; t; p, q)

∝∫

z∈Tn

1≤i<j≤n

Γp,q(tz±1i z±1

j )

Γp,q(z±1i z±1

j )

1≤i≤n

0≤r≤7 Γp,q(t1/2trz±1i )

Γp,q(z±2i )

,

where t2n+2t0t1t2t3t4t5t6t7 = p2q2.

Theorem:

II(n)

(t0, . . . , t7; t; p, q)

= II(n)

(t0u

,t1u

,t2u

,t3u

, ut4, ut5, ut6, ut7; t; p, q)

where u2 = pqtn+1

t4t5t6t7=

t0t1t2t3t4t5t6t7

.

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Page 22: Elliptic hypergeometric integrals - HCM: Hausdorff … · Elliptic hypergeometric integrals ... “elliptic Selberg integral”): Γp,q(t)n ... integral should give adjoint integral

Comments:

(1) Together with permutations of parameters,

generates group of order 2903040; Weyl group

E7! In fact, have partial E8 symmetry (dimen-

sion changes, must remain nonnegative inte-

ger)

(2) For t = q, gives solution to “elliptic Painleve

equation” (via a tau function). For t =√

q,

t = q2, obtain a new four-term bilinear recur-

rence with E8 symmetry. (Proof uses Plucker

relations for determinants and pfaffians respec-

tively)

(3) Multiplying the integrand by interpolation

functions gives generalization of transforma-

tion, indexed by one or two pairs of partitions.

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