0 A pr 2 00 7 Quantum Knizhnik – Zamolodchikov equation , Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices

AbstractWe present multiple-residue integral formulas for partial sums in the basis of link patterns of the polynomial solution of the level-1 $$U_q (\widehat{\mathfrak{s}\mathfrak{l}_2 })$$ quantum Knizhnik-Zamolodchikov equation at arbitrary values of the quantum parameter q. These formulas allow rewriting and generalizing a recent conjecture of Di Francesco connecting these sums to generating polynomials for weighted totally symmetric self-complementary plane partitions. We reduce the corresponding conjectures to a single integral identity, yet to be proved.

[1]  Greg Kuperberg,et al.  Symmetry classes of alternating-sign matrices under one roof , 2000 .

[2]  B. Lindström.  On the Vector Representations of Induced Matroids , 1973 .

[3]  D. Bressoud.  Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture , 1999 .

[4]  David P. Robbins,et al.  The Story of 1, 2, 7, 42, 429, 7436, … , 1991 .

[5]  J. Gier,et al.  The Quantum Symmetric Xxz Chain at ∆ = − , Alternating Sign Matrices and Plane Partitions , 2022 .

[6]  Quantum Incompressibility and Razumov Stroganov Type Conjectures , 2005, cond-mat/0506075.

[7]  Michio Jimbo,et al.  Algebraic Analysis of Solvable Lattice Models. , 1994 .

[8]  P. Zinn-Justin.  Combinatorial Point for Fused Loop Models , 2006, math-ph/0603018.

[9]  P. Zinn-Justin,et al.  From Orbital Varieties to Alternating Sign Matrices , 2005, math-ph/0512047.

[10]  Paul Zinn-Justin,et al.  Proof of the Razumov-Stroganov Conjecture for some Infinite Families of Link Patterns , 2006, Electron. J. Comb..

[11]  Doron Zeilberger,et al.  Proof of the alternating sign matrix conjecture , 1994, Electron. J. Comb..

[12]  N. Reshetikhin,et al.  Quantum affine algebras and holonomic difference equations , 1992 .

[13]  P. Zinn-Justin,et al.  LETTER TO THE EDITOR: The quantum Knizhnik Zamolodchikov equation, generalized Razumov Stroganov sum rules and extended Joseph polynomials , 2005, math-ph/0508059.

[14]  Paul Zinn-Justin,et al.  Around the Razumov-Stroganov Conjecture: Proof of a Multi-Parameter Sum Rule , 2004, Electron. J. Comb..

[15]  On Polynomials Interpolating Between the Stationary State of a O(n) Model and a Q.H.E. Ground State , 2006, cond-mat/0608160.

[16]  P. Zinn-Justin,et al.  Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Δ = −1/2 , 2007, 0704.3542.

[17]  B. Nienhuis,et al.  LETTER TO THE EDITOR: The quantum symmetric XXZ chain at Delta = - 1/2 , alternating-sign matrices and plane partitions , 2001 .

[18]  P. Francesco.  Quantum Knizhnik-Zamolodchikov equation and the weighted enumeration of Plane Partitions with symmetries , 2007 .

[19]  A. G. Izergin,et al.  Partition function of the six-vertex model in a finite volume , 1987 .

[20]  F. Smirnov.  A general formula for soliton form factors in the quantum sine-Gordon model , 1986 .

[21]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[22]  Bethe roots and refined enumeration of alternating-sign matrices , 2006, math-ph/0605004.

[23]  P. Francesco,et al.  Totally symmetric self-complementary plane partitions and the quantum Knizhnik–Zamolodchikov equation: a conjecture , 2006, cond-mat/0607499.

[25]  A. Razumov,et al.  Combinatorial Nature of the Ground-State Vector of the O(1) Loop Model , 2001 .