Elliptic hypergeometric

integrals

Eric M. Rains

Department of Mathematics

California Institute of Technology

MPIM Oberseminar, Bonn, 24/7/2008

Hypergeometric integrals

Gaussian integral:

ex

2/2dx =

2

(Hermite polynomials)

Gamma integral:

0x1exdx = ()

(Laguerre polynomials)

Beta integral:

1

0x1(1 x)1dx = ()()

( + )

(Jacobi polynomials)

1

Why hypergeometric?

1

0x1(1 x)1(1 tx)dx

=

0k

()( + k)( + k)

()( + + k)(1 + k)tk

=()()

( + )2F1(, ; + ; t)

Note transformations:

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

= (1 t)cab2F1(c b, c a; c; t)

= (1 t)b2F1(c a, b; c;t

1 t)

and evaluation

2F1(a, b; c; 1) =(c)(c a b)(c a)(c b)

2

Multivariate analogues

(Selberg) Let , , be complex numbers with

positive real parts. Then

1

n!

[0,1]n

1i 1. Then1

n!

Tn

1i

(Dirichlet) Let 0, . . . , n be complex numbers

with positive real parts. Then

xi=1

i

xi1i dxi =

i (i)

(

i)

When are real, a probability distribution (mul-

tivariate beta)

(Anderson; earlier by Varchenko, much earlier

by Dixon): Let 0, . . . , n be complex numbers

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

xi[ai+1,ai]

1i

Proof of Dirichlet integral: multiply by

u(

i)1 exp(u)du = (

i)

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

0i

Probabilistic interpretation of Dixon-Anderson

Integer i: take a Hermitian matrix with char-acteristic polynomial

i(ai)i and restrict toa random hyperplane; Dixon-Anderson is dis-

tribution of new eigenvalues.

Similarly for a real symmetric matrix with char-

acteristic polynomial

i( ai)2i.

Due to Baryshnikov in complex case with i 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 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.

6

Increasing subsequences

For a permutation Sn, an increasing sub-sequence is a subset S {1,2, . . . , n} with increasing on S. Let () be the maximum

size of an increasing subsequence of .

Theorem (Gessel, Rains) The number of Sn with () k is

EUU(k)|Tr(U)|2n.

If we choose n randomly (Poisson: probability

e2n/n!), then choose Sn uniformly atrandom, the probability that () k is

EUU(k) exp(2(Tr(U))).

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

= 1), with an extra factor

exp((zi +1/zi)).

7

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 = 1, al-

though symmetric models may have = 1/2,

= 2.

Many of these come from the Jack polynomial

identity

P(n) (1,1, . . . ,1; )

=

1i

q-analogues

Define

q(x) :=

0i(1 qix)1 =: 1/(x; q)

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

(Will use similar convention for other func-

tions)

Note

q(qx) = (1 x)q(x)

limq1

(1 q)aq(qa)(1 q)bq(qb)

=(a)

(b)

9

Macdonald-Morris integral:

1

n!

Tn

1i

Macdonald polynomials

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

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

1in xii ,

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

Macdonald conjectures:

1 Explicit formula for P(n) (1, t, . . . , t

n1; q, t).

2 Explicit formula for inner product.

3 Symmetry:

P(n) (. . . , q

itni, . . . ; q, t)

P(n) (. . . , t

ni, . . . ; q, t)is symmetric in , .

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

11

(Rahman)

1

2q(q)

zS1

0r4 q(trz1)q(Tz1, z2)

=

0r

Gustafsons proof

First, the identity:

1

q(q)n2nn!

zTn

1i

Elliptic analogues

Ruijsenaars elliptic Gamma function:

p,q(x) =

0j,k

1 pj+1qk+1/x1 pjqkx

Why elliptic? Consider

p(x) :=p,q(qx)

p,q(x)=

0k(1 pkx)(1 pk+1/x);

observe that

p(x) = xp(px),so p is a theta function on the elliptic curve

C/p.

Also note

p,q(px) = q(x)p,q(x)

p,q(x) = p,q(pq/x)1

0,q(x) = q(x).

14

Spiridonov (elliptic beta integral):

1

2p(p)q(q)

zS1

0r5 p,q(trz1)p,q(z2)

=

0r

Also, following Gustafson (elliptic Dixon-Anderson

integral):

1

p(p)nq(q)n2nn!

zTn

1i

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?

17

First key idea: double integral proof of Type II

integral should give adjoint integral operators.

Dixon-Anderson case ( = 1):

Evs((1 vv)A(1 vv)) s(A)

so Dixon-Anderson integral takes n1 variableSchur functions to n variable Schur functions.

(Similar to Okounkovs integral representation

for interpolation polynomials)

In general, elliptic Dixon-Anderson integral takes

n1 variable biorthogonal functions to n-variablebiorthogonal 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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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 Macdonalds 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?

19

Type II transformations

Define

II(n)

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

zTn

1i

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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