An Izergin-Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices
We obtain a new expression for the partition function of the 8VSOS model with domain wall boundary conditions, which we consider to be the natural extension of the Izergin-Korepin formula for the six-vertex model. As applications, we find dynamical (…
Authors: Hjalmar Rosengren
AN IZER GIN–K OREPIN-TYPE IDENTITY F OR THE 8VSOS MODEL, WITH APPLICA TIONS TO AL TERNA TING SIGN MA TRICE S HJALMAR ROSENGREN Abstract. W e obtain a new expression for the partition function of the 8VSOS model with domain wa ll b oundary conditions, which w e consider to be the natural extension of the Izergin– Korepin form ula for the six- v ertex mo del. As applications, we find dynamical (in the sense of the dynamical Y ang– Baxter equation) generalizations of the en umeration and 2-enumeration of alternating sign matrices. The dynamical en umeration has a n ice interpretation in terms of three-colourings of the square lattice. 1. Intr oduction An alternating sign matrix is a square matrix with en tries 0 , − 1 and 1, suc h that the non-zero en tries in each ro w and column form an alternating sequence of the f o rm 1 , − 1 , 1 , − 1 , . . . , − 1 , 1 . Mills, R o bbins and Rumsey [MRR] conjectured that the n um b er of n × n alter- nating sign mat r ices equals A n = 1! 4! 7! · · · (3 n − 2)! n !( n + 1)! · · · (2 n − 1 )! . This w as prov ed t hirteen years later b y Zeilb erger [Ze]. Kup erb erg [K1] found a simpler pro of based on the s ix-v ertex mo del on a square with domain w all b oundary conditions. This is a lattice mo del of statistical mechanic s, whose states can b e iden tified with alternating sign matr ices. Eac h state carries a w eigh t, in general dep ending on 2 n + 1 parameters q , x 1 , . . . , x n , y 1 , . . . , y n . The partition function f o r the mo del is the sum of the w eigh t of all states. By the Izergin– Korepin iden tit y [I, ICK], it can b e expres sed in terms of the determinant det 1 ≤ i,j ≤ n 1 ( x i − q y j )( x i − q − 1 y j ) . Kup erb erg observ ed that when q = e 2 π i/ 3 and x i = y i = 1 for all i , the w eigh t of eac h state can b e normalized to 1, so the partitio n function is equal to A n . The 1991 M athematics Subject Classific ation. 05A15, 82B20, 82B23. Key wor ds and phr ases. 8VSOS model, Izergin–Korepin identit y , partition function, alternating sign matrix. Researc h suppor ted by the Swedish Science Research Council (V etensk apsr ˚ adet). 1 2 HJALMAR ROSENGREN Izergin–Korepin iden tit y then giv es A n = 3 ( n +1 2 ) lim x 1 ,...,x n → 1 y 1 ,...,y n → 1 Y 1 ≤ i 0, η / ∈ Z + τ Z . W e will write p = e 2 π iτ and q = e 2 π iη . By q x w e alw a ys mean e 2 π iηx . W e will use the notation [ x ] = q − x/ 2 ∞ Y j =0 (1 − p j q x )(1 − p j +1 q − x ) . Up t o a m ultiplicativ e constan t, [ x ] equals the Jacobi theta function θ 1 ( η x | τ ) [WW]. W e sometimes write for short [ x 1 , . . . , x n ] = [ x 1 ] · · · [ x n ] . The function x 7→ [ x ] is o dd, en tire, and satisfies [ x + u, x − u , y + v , y − v ] − [ x + v , x − v , y + u, y − u ] = [ x + y , x − y , u + v , u − v ] . (2.1) In fact, up to an elemen tary multiplie r, the only suc h function is the Jacobi theta function, together with the degenerate cases [ x ] = sin ( π η x ) and [ x ] = x , cf. [WW, p. 461]. W e find it helpful to think of [ x ] as a t w o-parameter deformation of the n um b er x . In the case q N = 1 , w e find it more con v enien t to use the notation θ ( x ) = θ ( x ; p ) = ∞ Y j =0 (1 − p j x )(1 − p j +1 /x ) , θ ( x 1 , . . . , x n ) = θ ( x 1 , . . . , x n ; p ) = θ ( x 1 ; p ) · · · θ ( x n ; p ) , so t ha t [ x ] = q − x/ 2 θ ( q x ; p ) . The follo wing terminology will b e useful. Definition 2.1. Fixing τ an d η , we sa y that f i s a theta function of or der n and norm t if ther e exist c onstants a 1 , . . . , a n and C with a 1 + · · · + a n = t , such that f ( x ) = C [ x − a 1 ] · · · [ x − a n ] . (2.2) 4 HJALMAR ROSENGREN Equivalently, f i s an entir e function such that f ( x + 1 / η ) = ( − 1) n f ( x ) , f ( x + τ /η ) = ( − 1) n e 2 π iη ( t − nx ) − π iτ n f ( x ) . (2.3) The equiv alence of these tw o prop erties is classical [W e, p. 45]. More generally , an y function of the form f ( x ) = X j [ x − a ( j ) 1 ] · · · [ x − a ( j ) m + n ] [ x − b ( j ) 1 ] · · · [ x − b ( j ) m ] , where a ( j ) 1 + · · · + a ( j ) m + n − b ( j ) 1 − · · · − b ( j ) m = t for each j , satisfies the quasi-p erio dicity ( 2 .3). If f is en tire (that is, the singu- larities at x = b ( j ) i are all remo v able), it can then b e factored a s in (2.2). Unless f is iden tically zero, the zero set is then a i + Z η − 1 + Z τ η − 1 , 1 ≤ i ≤ n , where a 1 + · · · + a n = t . Th us, to prov e that f v anishes identically , it suffices to find n independen t zero es. This giv es a p o w erful (and classical) method for pro ving theta function identities , whic h we are going to apply rep eatedly . Finally , w e recall F rob enius’ determinan t ev aluation [F r] det 1 ≤ i,j ≤ n [ x i − y j + t ] [ x i − y j ] = ( − 1) ( n 2 ) [ t ] n − 1 | x | − | y | + t Q 1 ≤ i 0, and λ → ∞ . Up t o norma lizatio n, this limit corresp onds to the six-v ertex mo del. Since [ λ + a ] / [ λ + b ] → e iπ η ( b − a ) , w e o btain in the limit Z n ( x ; y ; ∞ ) = ( − 1) ( n 2 ) e iπ η ( | x | −| y | ) [1] n 2 [ γ ] n Q n i,j =1 [ x i − y j ][ x i + 1 − y j ] Q 1 ≤ i
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