Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Symmetry Classes of Alternating-Sign Matrices under One Roof

Greg Kuperberg
Annals of Mathematics
Second Series, Vol. 156, No. 3 (Nov., 2002), pp. 835-866
Published by: Annals of Mathematics
DOI: 10.2307/3597283
Stable URL: http://www.jstor.org/stable/3597283
Page Count: 32
  • Read Online (Free)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Symmetry Classes of Alternating-Sign Matrices under One Roof
Preview not available

Abstract

In a previous article [22], we derived the alternating-sign matrix (ASM) theorem from the Izergin-Korepin determinant [12], [19], [13] for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternating-sign matrices: VSASMs (vertically symmetric ASMs), even HTSASMs (half-turn-symmetric ASMs), and even QTSASMs (quarter-turn-symmetric ASMs). The VSASM enumeration was conjectured by Mills; the others by Robbins [30]. We introduce several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (two U-turn sides), OSASMs (off-diagonally symmetric ASMs), OOSASMs (off-diagonally, off-antidiagonally symmetric), and UOSASMs (off-diagonally symmetric with U-turn sides). UASMs generalize VSASMs, while UUASMs generalize VHSASMs (vertically and horizontally symmetric ASMs) and another new class, VHPASMs (vertically and horizontally perverse). OSASMs, OOSASMs, and UOSASMs are related to the remaining symmetry classes of ASMs, namely DSASMs (diagonally symmetric), DASASMs (diagonally, anti-diagonally symmetric), and TSASMs (totally symmetric ASMs). We enumerate several of these new classes, and we provide several 2-enumerations and 3-enumerations. Our main technical tool is a set of multi-parameter determinant and Pfaffian formulas generalizing the Izergin-Korepin determinant for ASMs and the Tsuchiya determinant for UASMs [37]. We evaluate specializations of the determinants and Pfaffians using the factor exhaustion method.

Page Thumbnails

  • Thumbnail: Page 
[835]
    [835]
  • Thumbnail: Page 
836
    836
  • Thumbnail: Page 
837
    837
  • Thumbnail: Page 
838
    838
  • Thumbnail: Page 
839
    839
  • Thumbnail: Page 
840
    840
  • Thumbnail: Page 
841
    841
  • Thumbnail: Page 
842
    842
  • Thumbnail: Page 
843
    843
  • Thumbnail: Page 
844
    844
  • Thumbnail: Page 
845
    845
  • Thumbnail: Page 
846
    846
  • Thumbnail: Page 
847
    847
  • Thumbnail: Page 
848
    848
  • Thumbnail: Page 
849
    849
  • Thumbnail: Page 
850
    850
  • Thumbnail: Page 
851
    851
  • Thumbnail: Page 
852
    852
  • Thumbnail: Page 
853
    853
  • Thumbnail: Page 
854
    854
  • Thumbnail: Page 
855
    855
  • Thumbnail: Page 
856
    856
  • Thumbnail: Page 
857
    857
  • Thumbnail: Page 
858
    858
  • Thumbnail: Page 
859
    859
  • Thumbnail: Page 
860
    860
  • Thumbnail: Page 
861
    861
  • Thumbnail: Page 
862
    862
  • Thumbnail: Page 
863
    863
  • Thumbnail: Page 
864
    864
  • Thumbnail: Page 
865
    865
  • Thumbnail: Page 
866
    866