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Symmetric Plane Partitions: From Robbins' Conjecture to ASM Proof

Explore the journey of Robbins' Conjecture to ASM proof, with contributions from Doron Zeilberger, Greg Kuperberg, and more, unraveling the mysteries of square ice and ASM's study by physicists.

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Symmetric Plane Partitions: From Robbins' Conjecture to ASM Proof

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  1. Totally Symmetric Self-Complementary Plane Partitions 1 0 0 –1 1 0 –1 0 1 1983

  2. Totally Symmetric Self-Complementary Plane Partitions 1 0 0 –1 1 0 –1 0 1

  3. 1 0 0 –1 1 0 –1 0 1

  4. Robbins’ Conjecture: The number of TSSCPP’s in a 2n X 2n X 2n box is 1 0 0 –1 1 0 –1 0 1

  5. Robbins’ Conjecture: The number of TSSCPP’s in a 2n X 2n X 2n box is 1 0 0 –1 1 0 –1 0 1 1989: William Doran shows equivalent to counting lattice paths 1990: John Stembridge represents the counting function as a Pfaffian (built on insights of Gordon and Okada) 1992: George Andrews evaluates the Pfaffian, proves Robbins’ Conjecture

  6. December, 1992 Doron Zeilberger (now at Rutgers University) announces a proof that # of ASM’s of size n equals of TSSCPP’s in box of size 2n. 1 0 0 –1 1 0 –1 0 1

  7. December, 1992 Doron Zeilberger (now at Rutgers University) announces a proof that # of ASM’s of size n equals of TSSCPP’s in box of size 2n. 1 0 0 –1 1 0 –1 0 1 1995 all gaps removed, published as “Proof of the Alternating Sign Matrix Conjecture,” Elect. J. of Combinatorics, 1996.

  8. Zeilberger’s proof is an 84-page tour de force, but it still left open the original conjecture: 1 0 0 –1 1 0 –1 0 1

  9. 1996 Greg Kuperberg (U.C. Davis) announces a simple proof 1 0 0 –1 1 0 –1 0 1 “Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices

  10. 1996 Greg Kuperberg (U.C. Davis) announces a simple proof 1 0 0 –1 1 0 –1 0 1 “Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices Physicists have been studying ASM’s for decades, only they call them square ice (aka the six-vertex model ).

  11. H O H O H O H O H O H H H H H H H O H O H O H O H O H H H H H H H O H O H O H O H O H H H H H H H O H O H O H O H O H H H H H H H O H O H O H O H O H 1 0 0 –1 1 0 –1 0 1

  12. 1960’s Rodney Baxter’s Triangle-to-triangle relation 1 0 0 –1 1 0 –1 0 1

  13. 1960’s Rodney Baxter’s Triangle-to-triangle relation 1 0 0 –1 1 0 –1 0 1 1980’s Vladimir Korepin Anatoli Izergin

  14. 1 0 0 –1 1 0 –1 0 1

  15. 1996 Doron Zeilberger uses this determinant to prove the original conjecture 1 0 0 –1 1 0 –1 0 1 “Proof of the refined alternating sign matrix conjecture,” New York Journal of Mathematics

  16. The End 1 0 0 –1 1 0 –1 0 1 (which is really just the beginning)

  17. Margaret Readdy Univ. of Kentucky The End 1 0 0 –1 1 0 –1 0 1 (which is really just the beginning)

  18. Margaret Readdy Univ. of Kentucky The End 1 0 0 –1 1 0 –1 0 1 Anne Schilling Univ. of Calif. Davis

  19. Margaret Readdy Univ. of Kentucky 1 0 0 –1 1 0 –1 0 1 Anne Schilling Univ. of Calif. Davis Mireille Bousquet-Mélou LaBRI, Univ. Bordeaux

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