Upload
trannguyet
View
229
Download
0
Embed Size (px)
Citation preview
Elliptic hypergeometric
integrals
Eric M. Rains
Department of Mathematics
California Institute of Technology
MPIM Oberseminar, Bonn, 24/7/2008
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)
1
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)
2
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)
3
(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
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.
5
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.
6
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
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)
8
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)
9
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.
10
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).
11
(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))
12
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!
13
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).
14
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.
15
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.
16
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 (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!)
18
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?
19
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
.
20
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.
21