Message from Richard Stanley about the Razumov-Stroganov Conjecture

From: Richard Stanley rstan at math dot mit dot edu
To: ayyer at physics.rutgers.edu, zeilberg at math.rutgers.edu
Subject: Razumov-Stroganov

Dear Professor Ayyer and Doron,

In your interesting approach toward the Razumov-Stroganov conjecture
you state that "To the best of our knowledge, this is the first time
the Razumov-Stroganov conjecture has been interpreted in this way."
I actually did something similar in a posting to the domino mailing
list in 2001 (appended below). At this time it seems that the complete
description of the eigenvector had not yet been conjectured.

Best regards,
Richard

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Date: Sat, 24 Feb 2001 16:57:06 -0500 (EST)
From: Richard Stanley 
To: propp@math.wisc.edu
CC: domino@math.wisc.edu
Subject: Re: the XXZ and 6V models

Jim asked:

> Can any of you explain what the XXZ model is about? 

Here is a restatement of the main conjecture of Razumov and Stroganov
in "familiar" terms:

Let n be a positive integer and N=2n+1. Let B be the set of binary
N-tuples u=(u_1,...,u_N) with n 1's, so #B = {N\choose n}. Regard u as
a circular sequence, so u_{N+1}=u_1. Define a real matrix P whose rows
and columns are indexed by B, as follows.

     P(u,u) = (N - 2a(u))/2, 

where a(u) is the number of entries u_i of u such that u_i = u_{i+1}.

If u \neq v, then P(u,v)=2 if u and v differ by an adjacent
transposition. Otherwise P(u,v)=0.

It is a theorem that P has an eigenvalue -3N/2. Let \psi be the
corresponding eigenvector. 

CONJECTURE. \psi may be chosen so that all entries are
positive. (Razumov-Stroganov state that this is a consequence of the
Perron-Frobenius theorem, though I don't immediately see this.)
Normalize \psi so that its least component is 1. Then its largest
component is A_n, the number of nxn alternating sign matrices.

NOTE. It's easy to see that \psi_u (the component of \psi indexed by
u) depends only on the cyclic equivalence class (necklace) of u. 

EXAMPLE. n=2. Order the elements of B as
11000,01100,00110,00011,10001,10100,01010,00101,10010,01001,
The matrix P has the block form [X Y]
                                [Z W]
where each block is 5x5. X is the diagonal matrix with -1/2 on the
diagonal. Y is the circulant matrix with first row (2,0,0,0,2). Z is
the circulant matrix with first row (2,2,0,0,0). W is the circulant
matrix with first row (3/2,0,2,2,0).  -15/2 is an eigenvalue, with
eigenvector \psi = (1,1,1,1,1,2,2,2,2,2).

Richard