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Identifier
Values
=>
Cc0005;cc-rep
[1,0]=>0 [1,0,1,0]=>1 [1,1,0,0]=>0 [1,0,1,0,1,0]=>2 [1,0,1,1,0,0]=>0 [1,1,0,0,1,0]=>0 [1,1,0,1,0,0]=>1 [1,1,1,0,0,0]=>0 [1,0,1,0,1,0,1,0]=>3 [1,0,1,0,1,1,0,0]=>0 [1,0,1,1,0,0,1,0]=>0 [1,0,1,1,0,1,0,0]=>1 [1,0,1,1,1,0,0,0]=>0 [1,1,0,0,1,0,1,0]=>0 [1,1,0,0,1,1,0,0]=>0 [1,1,0,1,0,0,1,0]=>1 [1,1,0,1,0,1,0,0]=>1 [1,1,0,1,1,0,0,0]=>0 [1,1,1,0,0,0,1,0]=>0 [1,1,1,0,0,1,0,0]=>0 [1,1,1,0,1,0,0,0]=>1 [1,1,1,1,0,0,0,0]=>0 [1,0,1,0,1,0,1,0,1,0]=>4 [1,0,1,0,1,0,1,1,0,0]=>0 [1,0,1,0,1,1,0,0,1,0]=>0 [1,0,1,0,1,1,0,1,0,0]=>1 [1,0,1,0,1,1,1,0,0,0]=>0 [1,0,1,1,0,0,1,0,1,0]=>0 [1,0,1,1,0,0,1,1,0,0]=>0 [1,0,1,1,0,1,0,0,1,0]=>2 [1,0,1,1,0,1,0,1,0,0]=>1 [1,0,1,1,0,1,1,0,0,0]=>0 [1,0,1,1,1,0,0,0,1,0]=>0 [1,0,1,1,1,0,0,1,0,0]=>0 [1,0,1,1,1,0,1,0,0,0]=>1 [1,0,1,1,1,1,0,0,0,0]=>0 [1,1,0,0,1,0,1,0,1,0]=>0 [1,1,0,0,1,0,1,1,0,0]=>0 [1,1,0,0,1,1,0,0,1,0]=>0 [1,1,0,0,1,1,0,1,0,0]=>0 [1,1,0,0,1,1,1,0,0,0]=>0 [1,1,0,1,0,0,1,0,1,0]=>1 [1,1,0,1,0,0,1,1,0,0]=>0 [1,1,0,1,0,1,0,0,1,0]=>1 [1,1,0,1,0,1,0,1,0,0]=>2 [1,1,0,1,0,1,1,0,0,0]=>0 [1,1,0,1,1,0,0,0,1,0]=>0 [1,1,0,1,1,0,0,1,0,0]=>0 [1,1,0,1,1,0,1,0,0,0]=>1 [1,1,0,1,1,1,0,0,0,0]=>0 [1,1,1,0,0,0,1,0,1,0]=>0 [1,1,1,0,0,0,1,1,0,0]=>0 [1,1,1,0,0,1,0,0,1,0]=>0 [1,1,1,0,0,1,0,1,0,0]=>0 [1,1,1,0,0,1,1,0,0,0]=>0 [1,1,1,0,1,0,0,0,1,0]=>1 [1,1,1,0,1,0,0,1,0,0]=>1 [1,1,1,0,1,0,1,0,0,0]=>1 [1,1,1,0,1,1,0,0,0,0]=>0 [1,1,1,1,0,0,0,0,1,0]=>0 [1,1,1,1,0,0,0,1,0,0]=>0 [1,1,1,1,0,0,1,0,0,0]=>0 [1,1,1,1,0,1,0,0,0,0]=>1 [1,1,1,1,1,0,0,0,0,0]=>0
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Description
The maximal $n$ such that the minimal generator-cogenerator module in the linear Nakayama algebra corresponding to a Dyck path is $n$-rigid.
The correspondence between linear Nakayama algebras and Dyck paths is explained in St000684The global dimension of the LNakayama algebra associated to a Dyck path. and on the Nakayama algebras page. A module $M$ is $n$-rigid if $\operatorname{Ext}^i(M,M)=0$ for $1\le i\le n$.
This statistic is the maximal $n$ such that the minimal generator-cogenerator module $A\oplus D(A)$ of the linear Nakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
References
[1] Marczinzik, R. Upper bounds for the dominant dimension of Nakayama and related algebras arXiv:1605.09634
[2] Iyama, O. Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories MathSciNet:2298819 arXiv:math/0407052
Code
gap('LoadPackage("QPA");')

import tempfile as _tf, os as _os
_gap_code = r"""
DeclareOperation("domdimend", [IsList]);

InstallMethod(domdimend, "for a representation of a quiver", [IsList],0,function(L)
    local A, CoRegA, M, N, R, RegA, W, injA, list, n, projA;
    list := L;
    A := NakayamaAlgebra(GF(3),list);
    R := [0..20];
    projA := IndecProjectiveModules(A);
    RegA := DirectSumOfQPAModules(projA);
    injA := IndecInjectiveModules(A);
    CoRegA := DirectSumOfQPAModules(injA);
    N := DirectSumOfQPAModules([RegA,CoRegA]);
    M := BasicVersionOfModule(N);
    W := Filtered(R,x->N_RigidModule(M,x)=true);
    n := Maximum(W);
    return(n);
end
);
"""
with _tf.NamedTemporaryFile(mode="w", suffix=".g", delete=False, dir="/tmp") as _f:
    _f.write('LoadPackage("QPA");;\n')
    _f.write(_gap_code)
    _tmp = _f.name
gap.eval('Read("' + _tmp + '");')
_os.unlink(_tmp)

def kupisch(D):
    DR = D.reverse()
    H = DR.heights()
    return [1 + H[i] for i, s in enumerate(DR) if s == 0] + [1]

def statistic(D):
    K = kupisch(D)
    return ZZ(gap.domdimend(K))
Created
Jan 18, 2017 at 00:26 by Rene Marczinzik
Updated
Mar 11, 2026 at 17:55 by Nupur Jain