Sketch of a Proof of an Intriguing Conjecture of Karola Mészáros and Alejandro Morales

Regarding the Volume of the D_{n} Analog of the Chan-Robbins-Yuen Polytope (Or: The Morris-Selberg Constant Term Identity Strikes Again!)

By Doron ZEILBERGER

To Dick Askey (b. June 4, 1933): from a 4^{3}-year-old to a 3^{4}-year-old, and thanks

for preaching the importance of Constant Term Identities!

Disclaimer: Some of the steps below (in particular, the “change of variable” in contour-integrals) require ‘rigorous’ justification, that I am sure could be easily supplied by a skilled analyst.

Note added July 14, 2014: Jang Soo Kim independently found the same proof, but his proof

is complete! See http://arxiv.org/abs/1407.3467.

Recall that for any rational function f(z) of a variable z, CT_{z}f(z) is the coeff. of z^{0} in the formal Laurent expansion of f(z) (that always exists!).

Karola Mészáros and Alejandro Morales have recently made the following intriguing conjecture.

Conjecture ([MeMora], Conj. 7.12, also presented by Morales[Mora] at Stanley@70)

CT_{x_{n}} CT_{x_{n−1}} … CT_{x_{1}} ∏_{i=1}^{n} x_{i}^{−1}(1−x_{i})^{−2} ∏_{1≤i<j≤n} (x_{j} −x_{i})^{−1}(1−x_{j} −x_{i})^{−1} = 2^{n^{2}} ∏_{k=1}^{n} Cat(k) ,

where Cat(k) are the ubiquitous Catalan numbers (2k)!/(k!(k + 1)!) (that Igor Pak believes are better than primes for searching for ETI!)

The present conjecture is a D_{n}-analog of a conjecture made in [CRY], and proved in [Z], using the versatile Morris-Selberg Constant Term Identity ([Morr], restated in [Z]):

CT_{x_{n}} … CT_{x_{1}} ∏_{i=1}^{n}(1 − x_{i})^{−a} ∏_{i=1}^{n} x_{i}^{−b} ∏_{1≤i<j≤n}(x_{j} − x_{i})^{−2c} = S_{n}(a, b, c) ,  (Chip)

where

S_{n}(a, b, c) := \frac{1}{n!} ∏_{j=0}^{n−1} \frac{Γ(a + b + (n − 1 + j)c)Γ(c)}{Γ(a + jc)Γ(c + jc)Γ(b + jc + 1)} .

By using Cauchy’s theorem, this is equivalent to

(\frac{1}{2πi})^{n} ∫_{C} ∏_{i=1}^{n}(1 − x_{i})^{−a} ∏_{i=1}^{n} x_{i}^{−b−1} ∏_{1≤i<j≤n}(x_{j} − x_{i})^{−2c} ∏_{i=1}^{n} d x_{i} = S_{n}(a, b, c) ,  (Atle)

where now a, b, c can be any real numbers (with obvious conditions to ensure convergence) and C is any multi-contour in n-dimensional complex space, that is far enough from the origin.

The Mészáros-Morales conjecture is the special case a = 2, c = \frac{1}{2} of the following fact.

Fact : Let a be a positive integer, and c a positive half-integer, then

CT_{x_{n}} CT_{x_{n−1}} … CT_{x_{1}} ∏_{i=1}^{n} x_{i}^{−(a−1)}(1 − x_{i})^{−a} ∏_{1≤i<j≤n} (x_{j} − x_{i})^{−2c}(1 − x_{j} − x_{i})^{−2c}

= 2^{2cn(n−1)+2(a−1)n} · S_{n}(a , −\frac{1}{2} , c) .

Proof: Converting the iterated constant-terms to a multi-contour-integral, we have to evaluate

(\frac{1}{2πi})^{n} ∫_{C} ∏_{i=1}^{n} x_{i}^{−a}(1 − x_{i})^{−a} ∏_{1≤i<j≤n} (x_{j} − x_{i})^{−2c}(1 − x_{j} − x_{i})^{−2c} ∏_{i=1}^{n} d x_{i} .

Now make the change of variables:

x_{i} = \frac{1 − z_{i}}{2} , (1 ≤ i ≤ n) ,

getting that our multi-integral equals

(\frac{1}{2πi})^{n} · 2^{2an} · 2^{2cn(n−1)}(−\frac{1}{2})^{n} ∫_{C′} ∏_{i=1}^{n}(1 − z_{i})^{−a}(1 + z_{i})^{−a} ∏_{1≤i<j≤n} (z_{i} − z_{j})^{−2c}(z_{i} + z_{j})^{−2c} ∏_{i=1}^{n} d z_{i}

= (\frac{1}{2πi})^{n} · (−1)^{n}2^{2an+2cn(n−1)−n} ∫_{C′} ∏_{i=1}^{n}(1 − z_{i}^{2})^{−a} ∏_{1≤i<j≤n} (z_{i}^{2} − z_{j}^{2})^{−2c} ∏_{i=1}^{n} d z_{i} ,

where C′ is some other multi-contour. Now is time for yet another change of variable

w_{i} = z_{i}^{2} , (1 ≤ i ≤ n) ,

and we have

∏_{i=1}^{n} d z_{i} = (\frac{1}{2})^{n} ∏_{i=1}^{n} \frac{d w_{i}}{w_{i}^{1/2}} ,

getting that our multi-integral is

(−1)^{n+n(n−1)c}(\frac{1}{2πi})^{n} 2^{2an+2cn(n−1)−2n} ∫_{C′′} ∏_{i=1}^{n}(1−w_{i})^{−a} ∏_{i=1}^{n} w_{i}^{−1/2} ∏_{1≤i<j≤n} (w_{j} −w_{i})^{−2c} ∏_{i=1}^{n} d w_{i} ,

for yet another multi-contour, C′′, and thanks to (Atle) this is

2^{2cn(n−1)+2(a−1)n} · S_{n}(a , −\frac{1}{2} , c) ,

times (−1)^{n+n(n−1)c}, and I am sure that this annoying extra sign could be explained away.

References

[CRY] Clara S. Chan, David P. Robbins, and David S. Yuen, On the volume of a certain polytope, Experimental Mathematics 9(2000), 91-99. http://arxiv.org/abs/math/9810154 .

[MeMora] Karola Mészáros and Alejandro H. Morales, Flow polytopes of signed graphs and the Kostant partition function, http://arxiv.org/abs/1208.0140, to appear in International Mathematics Research Notices.

[Mora] Alejandro H. Morales, Open Problems session, Stanley@70, June 23, 2014, ca. 4:50-5:00pm,

http://math.mit.edu/stanley70/Site/Program.html .
Slides are available here: http://math.mit.edu/stanley70/Site/Slides/Morales.pdf

[Morr] William (“Chip”) G. Morris, “Constant Term Identities for Finite and Affine Root Systems: Conjectures and Theorems”, PhD thesis, University of Wisconsin-Madison, 1982. [Advisor: Richard Askey].

[Z] Doron Zeilberger, Proof of a Conjecture of Chan, Robbins, and Yuen, Elec. Trans. of Numerical Analysis (ETNA), 9(1999), 147-148.

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/cry.html.

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

url: http://www.math.rutgers.edu/~zeilberg/ .
Email: zeilberg at math dot rutgers dot edu .

EXCLUSIVELY PUBLISHED IN THE PERSONAL JOURNAL OF SHALOSH B.

EKHAD and DORON ZEILBERGER http://www.math.rutgers.edu/~zeilberg/pj.html
and arxiv.org .

July 9, 2014