"Trivializing" Generalizations of some Izergin-Korepin-type Determinants

We generalize (and hence trivialize and routinize) numerous explicit evaluations of determinants and pfaffians due to Kuperberg, as well as a determinant of Tsuchiya. The level of generality of our statements render their proofs easy and routine, by using Dodgson Condensation and/or Krattenthaler’s factor exhaustion method. All our matrices will be assumed to be embedded inside an infinite matrix. The first theorem adds parameters to the determinant formulas found in Kuperberg [Ku] (Theorem 15), as well as older determinants, mentioned there, due to Cauchy, Stembridge, Laksov-Lascoux-Thorup, and Tsuchiya [T]. This way, the formulation is suited to the method of [AZ]. Our proofs are much more succinct and automatable, since their generality enables an easy induction using Dodgson’s rule [D, AZ], or by employing Krattenthaler’s elegant factor exhaustion method [Kr1]. Relevant background for this paper can found in [Ku], and references thereof. Theorem 1: det ( 1 xi + yj +Axiyj )1,n i,j = ∑ ii γ(p j−i, yj−i)2γ(qpj−i, xyj−i)γ(xpj−i, qyj−i) ∏ i,j γ(pn+j−i, yn+j−i) . det ( τ(qj−i, xj−i) τ(pj−i, yj−i) )1,n i,j = ∏ 2|j−i>0 γ(p j−i, yj−i)2 ∏ 2-j−i>0 γ(qp j−i, xyj−i)γ(xpj−i, qyj−i) (qx)bn/4c ∏ j>i τ(pj−i, yj−i)2 . det(Z1) 1,n i,j = δe,n γ(q, x)(py) (qx)bn/4c ∏ 2|j−i>0 γ(p j−i, yj−i)2γ(qpj−i, xyj−i)γ(xpj−i, qyj−i) ∏ j>i τ(pj−i, yj−i)2 . det(Z2) 1,n i,j = γ(q, x) (qx)b(n−1)/4c ∏ 2|j−i>0 γ(p j−i, yj−i)2γ(qpj−i, xyj−i)γ(xpj−i, qyj−i) τ(pn, yn)1−δe,n ∏ j>i τ(pj−i, yj−i)2 . det(Z3) 1,n i,j = (−py)γ(q, x) (qx)b(n+1)/4c ∏ 2|j−i>0 γ(p j−i, yj−i)2γ(qpj−i, xyj−i)γ(xpj−i, qyj−i) τ(pn, yn)1−δe,n ∏ j>i τ(pj−i, yj−i)2 . “TRIVIALIZING” GENERALIZATIONS OF SOME IZERGIN-KOREPIN-TYPE DETERMINANTS 3 det(Z4) 1,n i,j = 2 n−1 q na + (−1)x (qx)n(n−1)(n+1−3a)/6 ∏ j 6=i γ(q |j−i|, x|j−i|) ∏ i,j τ(qa+j−i, xa+j−i) . det(Z5) 1,n i,j = (−1)( n 2)2n−1 q + x (qx)n(n−1)(n+1−3b)/6 ∏ j 6=i γ(q |j−i|, x|j−i|) ∏ i,j γ(qb+j−i, xb+j−i) . Pf =det ( λi,j γ(qj−i, xj−i)γ(rj−i, zj−i) γ(pj−i, yj−i) )1,2n i,j = (yp) (qxrz)n ∏1,n j 6=i γ(p |j−i|, y|j−i|)2 ∏1,n i,j γ(pn+j−i, yn+j−i)2 × 1,n∏i,jγ(qp|j−i|, xy|j−i|)γ(xp|j−i|, qy|j−i|)γ(rp|j−i|, zy|j−i|)γ(zp|j−i|, ry|j−i|).Sketch of Proof: Identities Z4 and Z5 are directly amenable to Dodgson‘s Condensation technique[AZ]. For the remaining assertions, use the factor exhaustion method [Kr1] (see also [Ku]): the essentialidea is to compare zeros and poles on both sides of the equation at hand. We leave the straightforwarddetails to the reader.Acknowledgment: The authors wish to thank Christian Krattenthaler for helpful suggestions. References[AZ] T. Amdeberhan, D. Zeilberger, Determinants Through the Looking Glass, Adv. Appl. Math. (Foata special issue)2/3 27 (2001), 225-230.[D] C.L. Dodgson, Condensation of Determinants, Proceedings of the Royal Society of London 15 (1866), 150-155.[Kr1] C. Krattenthaler, Advanced determinant calculus, Sèminaire Lotharingien Combin. (B42q) 42 (1999).[Kr2] C. Krattenthaler, Advanced determinant calculus: A Complement, Linear Alg. Appl. 411 (2005), 68-166.[Ku] G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Ann. of Math. (2) 156 (2002),835-866.[T] O. Tsuchiya, Determinant formula for the six-vertex model with reflecting end, J. Math. Phys. 39 (1998), 5946-5951.