Alternating sign matrix theorem

introduction

• PDF
• descending plane partitions and alternating sign matrix  [1]http://math.berkeley.edu/~reshetik/RTG-semin-fall-2010/Philippe.pdf[2]
• Refined enumeration of Alternating Sign Matrices and Descending Plane Partitions

 

 

 

 

 

 

 

 

DPP

• http://mathworld.wolfram.com/DescendingPlanePartition.html
• number of DPPs with parts at most n is given by Andrews in 1979.
• number of ASM of size n is same as the above sequence

 

 

DPP to lattice paths

• P. Lalonde, Lattice paths and the antiautomorphism of the poset of descending plane partitions, Discrete Math. 271 (2003) 311–319
• Descending plane partitions and rhombus tilings of a hexagon with a triangular hole C. Krattenthaler, 2006
• Rhombus tilings/Dimers or Lattice Paths for DPPs
• lattice paths (lattice fermions)
• related to non-intersecting paths
• Gessel-Viennot theorem

 

 

 

 

from ASM to 6 vertex model with domain wall boundary condition(6VDW)

• Kuperberg
• Izergin - Korepin

 

 

1+1 dimensional Lorentzian quantum gravity

exists quantities \phi such that if \phi(g,a)=\phi'(g',a') then [T(a,g),T(a',g')]=0

\phi(g,a)=\frac{1-g^2(1-a^2)}{ag}=q+q^{-1}

 

 

 

history

• 1983 Mills, Robbins and Rumsey ASM conjecture
• 198? Korepin recurrence relation for 6VDW
• 1987 Izergin. determinant function of the partition function of the 6VDW based on Korepin's work
• 1996 Zilberger proof of ASM conjecture
• 1996 Kuperberg alternative proof of ASM conjecture using the connection with the six vertex model
• 2011 correspondence between DPP and ASM
• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxTVJuMk9keXA4cEE/edit

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Plane_partition
• http://en.wikipedia.org/wiki/alternating_sign_matrix
• http://en.wikipedia.org/wiki/Six-vertex_model

 

 

 

books

• 찾아볼 수학책
• R. J. Baxter Exactly Solved Models in Statistical mechanics
• Proofs and Confirmations
  • Bressoud, David M.,
  • MAA Spectrum, Mathematical Associations of America, Washington, D.C., 1999.
  • Proofs and Confirmation

 

expositions

• http://www.macalester.edu/~bressoud/talks/
• http://www.macalester.edu/~bressoud/talks/2009/asm-Moravian.pdf

 

 

articles

• http://www.math.lsa.umich.edu/~lserrano/asm.pdf
• Propp, James. 2002. The many faces of alternating-sign matrices. math/0208125 (August 15). http://arxiv.org/abs/math/0208125. 
• How the alternating sign matrix conjecture was solved,
  
  • Bressoud, David M. and Propp, James,
  • Notices of the American Mathematical Society, 46 (1999), 637-646.
• Another proof of the alternating sign matrix conjecture
  
  • G Kuperberg, International Mathematics Research Notes (1996), 139-150.
• Proof of the alternating sign matrix conjecture
  
  • Zeilberger, Doron
  • Electronic Journal of Combinatorics 3 (1996), R13.
• Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase
  
  • Bleher, Pavel M.; Fokin, Vladimir V.