

A029729


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


2



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

1,2


COMMENTS

Also, ratio of vector elements of the ground state in the loop representation of the braidmonoid 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 noncrossing chord diagram, divided by the overall smallest element. We reduce the standard braidmonoid 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. ZinnJustin and A. Knutson and P. ZinnJustin.
These numbers arise in a similar way to A005130 and related sequences appear in the groundstate of the integrable TemperleyLieb 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 ZinnJustin, Mar 05 2023


LINKS

Paul ZinnJustin, Table of n, a(n) for n = 1..16
Jan de Gier, Loops, matchings and alternatingsign matrices, arXiv:math/0211285 [math.CO], 20022003.
P. Di Francesco and P. ZinnJustin, Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties, Comm. Math. Phys., 262(2):459487, 2006; arXiv preprint, arXiv:mathph/0412031, 20042005.
A. Garbali and P. ZinnJustin, Shuffle algebras, lattice paths and the commuting scheme, arXiv:2110.07155 [math.RT], 20212022. See also Macaulay2 code to generate the sequence.
A. Knutson and P. ZinnJustin, A scheme related to the Brauer loop model, Adv. Math., 214(1):4077, 2007, arXiv preprint, arXiv:math/0503224 [math.AG], 20052006.
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:condmat/9709051 [condmat.statmech], 1997; Phys. Rev. Lett. Vol. 81 (1998) pp. 504507.


FORMULA

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


EXAMPLE

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 CohenMacaulay 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


CROSSREFS

Cf. A005130.
Sequence in context: A219266 A022514 A094579 * A319253 A328811 A136584
Adjacent sequences: A029726 A029727 A029728 * A029730 A029731 A029732


KEYWORD

nonn,nice


AUTHOR

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


EXTENSIONS

Entry revised based on comments from Paul ZinnJustin, Mar 14 2005
Terms a(12) and beyond from Paul ZinnJustin, Mar 05 2023


STATUS

approved



