A029729 Degree of the variety of pairs of commuting n X n matrices.

1, 3, 31, 1145, 154881, 77899563, 147226330175, 1053765855157617, 28736455088578690945, 3000127124463666294963283, 1203831304687539089648950490463, 1862632561783036151478238040096092649, 11143500837236042423379349834982088594105985

Offset: 1

Author

Nolan R. Wallach (nwallach(AT)euclid.ucsd.edu), Dec 11 1999

Keywords

• nonn
• nice

Comments

Also, ratio of vector elements of the ground state in the loop representation of the braid-monoid Hamiltonian H = Sum_i (3 - 2 e_i - b_i) with size 2n and periodic boundary conditions. Specifically the smallest element that corresponds to a non-crossing chord diagram, divided by the overall smallest element. We reduce the standard braid-monoid algebra to the Brauer algebra B_{2n}(1). - B. Nienhuis & J. de Gier (B.Nienhuis(AT)UvA.NL), May 13 2004. For a proof that this is the same sequence, see the articles by P. Di Francesco and P. Zinn-Justin and A. Knutson and P. Zinn-Justin.

These numbers arise in a similar way to A005130 and related sequences appear in the groundstate of the integrable Temperley-Lieb Hamiltonian.

It is also the weighted enumeration of lattice paths on an n X n square lattice going from the left side to the top side, with same initial and final orders of paths, and with a weight of 2 per vertex where a path turns 90 degrees. - Paul Zinn-Justin, Mar 05 2023

Examples

			n=1: Degree of C X C which is 1. n=2: The degree can be calculated by hand to be 3. n=3: See Macaulay manual (link above): one of steps in proof that variety for 3 X 3 is Cohen-Macaulay is to compute the degree which is 31. (n=4) Matt Clegg (CS at UCSD) and Nolan Wallach using 10 Sun Workstations and a distributed Grobner Basis package (in 1993).
(2(e1 + e2 + e3 + e4) + b1 + b2 + b3 + b4)(G + G e2 + b2)(e1 e3 b2) = 12 (G + G e2 + b2)(e1 e3 b2) with G = 3, therefore a(2) = 3
		

Links

• Paul Zinn-Justin, Table of n, a(n) for n = 1..16
• Jan de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.
• P. Di Francesco and P. Zinn-Justin, Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties, Comm. Math. Phys., 262(2):459-487, 2006; arXiv preprint, arXiv:math-ph/0412031, 2004-2005.
• A. Garbali and P. Zinn-Justin, Shuffle algebras, lattice paths and the commuting scheme, arXiv:2110.07155 [math.RT], 2021-2022. See also Macaulay2 code to generate the sequence.
• A. Knutson and P. Zinn-Justin, A scheme related to the Brauer loop model, Adv. Math., 214(1):40-77, 2007, arXiv preprint, arXiv:math/0503224 [math.AG], 2005-2006.
• Macaulay 2 Manual, Test of matrix routines, Viewed May 03 2016.
• M. J. Martins, B. Nienhuis, and R. Rietman, An Intersecting Loop Model as a Solvable Super Spin Chain, arXiv:cond-mat/9709051 [cond-mat.stat-mech], 1997; Phys. Rev. Lett. Vol. 81 (1998) pp. 504-507.
• Ada Stelzer and Alexander Yong, Combinatorial commutative algebra rules, arXiv:2306.00737 [math.CO], 2023.

Crossrefs

Cf. A005130.

Formula

There is a formula in terms of divided differences operators (too complicated to reproduce here).

Extensions

Entry revised based on comments from Paul Zinn-Justin, Mar 14 2005

Terms a(12) and beyond from Paul Zinn-Justin, Mar 05 2023