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Gröbner bases of ideals invariant under a commutative group

2013
*
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13
*

We propose efficient algorithms to compute

doi:10.1145/2465506.2465944
dblp:conf/issac/FaugereS13
fatcat:mej5g5bxmzhwxoj67vqta5zgeu
*the**Gröbner*basis*of*an*ideal*I ⊂ k[x 1 , . . . , x n ] globally*invariant**under**the*action*of**a**commutative*matrix*group*G, in*the**non*-*modular**case*(where char ...*The*idea is to simultaneously diagonalize*the*matrices in G, and apply*a*linear change*of*variables on I corresponding to*the**base*-change matrix*of*this diagonalization. ... EXPERIMENTS In this section, we report some experiments that show*the*improvements given by our approach on*the*computation*of**Gröbner*basis*of**ideals**invariant**under**a**commutative**group*. ...##
###
Calculating Invariant Rings of Finite Groups over Arbitrary Fields

1996
*
Journal of symbolic computation
*

Up to now, such algorithms have been available only for

doi:10.1006/jsco.1996.0017
fatcat:4qx3ifrlkned3frja4mljenmby
*the**case*that*the*characteristic*of*K does not divide*the**group*order. ... Some applications to*the*question whether*a**modular**invariant*ring is Cohen-Macaulay or isomorphic to*a*polynomial ring are discussed. ... Matzat, who raised my interest in*invariant*theory, and to*the*referees for their valuable comments on*the*first version*of*this paper. ...##
###
The isomorphism problem for graded algebras and its application to mod-p cohomology rings of small p-groups
[article]

2015
*
arXiv
*
pre-print

*The*mod-p cohomology ring

*of*

*a*

*non*-trivial finite p-

*group*is an infinite dimensional, finitely presented graded unital algebra over

*the*field with p elements, with generators in positive degrees. ... As application, we determine all graded isomorphisms between

*the*mod-p cohomology rings

*of*all p-

*groups*

*of*order at most 100. ...

*Based*on this,

*the*nilradicals

*of*

*modular*cohomology rings

*of*finite

*groups*can be computed by intersecting

*the*preimages

*of*certain explicitly given

*ideals*

*under*morphisms (namely restrictions)

*of*finitely ...

##
###
Bifurcating entanglement-renormalization group flows of fracton stabilizer models
[article]

2019
*
arXiv
*
pre-print

We calculate bifurcating entanglement-renormalization

arXiv:1909.12304v1
fatcat:unz7muuyubamvepz3yfuewsose
*group*flows for*a*wide range*of*examples and,*based*on those results, propose conjectures regarding*the*classification*of*translation-*invariant*topological ... We investigate*the*entanglement-renormalization*group*flows*of*translation-*invariant*topological stabilizer models in three dimensions. ... This work is supported by start-up funds at Yale University (DW and MC), NSF*under*award number DMR-1846109 and*the*Alfred P. Sloan foundation (MC). ...##
###
Rings of invariants for modular representations of elementary abelian p-groups

2013
*
Transformation groups
*

We initiate

doi:10.1007/s00031-013-9207-z
fatcat:rta7t5jrzzcmpfdzs33fdothra
*a*study*of**the*rings*of**invariants**of**modular*representations*of*elementary abelian p-*groups*. ... With*a*few notable exceptions,*the**modular*representation theory*of*an elementary abelian p-*group*is wild. However, for*a*given dimension, it is possible to parameterise*the*representations. ...*A*SAGBI basis is*the*Subalgebra Analogue*of**a**Gröbner*Basis for*Ideals*. ...##
###
Bifurcating entanglement-renormalization group flows of fracton stabilizer models

2020
*
Physical Review Research
*

We calculate bifurcating entanglement-renormalization

doi:10.1103/physrevresearch.2.033021
fatcat:tnawxsfg7nho3ho6rgsw4j35zy
*group*flows for*a*wide range*of*examples and,*based*on those results, propose conjectures regarding*the*classification*of*translation-*invariant*topological ... We investigate*the*entanglement-renormalization*group*flows*of*translation-*invariant*topological stabilizer models in three dimensions. ... For*the**ideal*J x*a*= (z(y 2 + y + 1), y 2 l + b, z 2 l + c), we have*the*following*Gröbner**bases*. (1) Suppose b 2 + b + 1 = 0 and c = 0. Then we have*the*following*cases*. (*a*) Suppose 3 2 l + 1. ...##
###
Computing Modular Invariants of p-groups

2002
*
Journal of symbolic computation
*

We show that there exists

doi:10.1006/jsco.2002.0558
fatcat:zcrkmcs7mzee7huedtjnnfwpmm
*a*choice*of*basis and monomial order for which*the*ring*of**invariants*, k[V ] G , has*a*finite SAGBI basis. ... Let V be*a*finite dimensional representation*of**a*p-*group*, G, over*a*field, k,*of*characteristic p. ... Acknowledgements We thank Gregor Kemper for reading an earlier draft*of*this paper and suggesting some improvements and corrections. ...##
###
Page 4972 of Mathematical Reviews Vol. , Issue 2003g
[page]

2003
*
Mathematical Reviews
*

They prove many

*cases**of*when*the*CS test*ideal*extends*under**a**base*change, which then enables them to prove various*cases**of*tightly closed*ideals*. ...*The*authors present*a*simple proof for Noether’s bound in*the**non*-*modular**case*, which is due to Fleischmann, R. Fogarty and D. Benson. ...##
###
NOETHER NUMBERS FOR SUBREPRESENTATIONS OF CYCLIC GROUPS OF PRIME ORDER

2002
*
Bulletin of the London Mathematical Society
*

*The*maximum degree

*of*an indecomposable element

*of*

*the*algebra

*of*

*invariants*, k[W ] Z/p , is called

*the*Noether number

*of*

*the*representation, and is denoted by β(W ). ... We shall also give

*a*lower bound for β(V ) for any

*modular*representation

*of*Z/p, and we compute

*a*SAGBI basis for k[V 2 ⊕ V 3 ] Z/p , where k is ... We would like to thank Gregor Kemper for providing us with

*a*closed form for

*the*Hilbert series, P(k[V 2 ⊕ V 3 ] Z/p , λ), which we required for our proof

*of*Theorem 5.1. ...

##
###
Rings of invariants for modular representations of elementary abelian p-groups
[article]

2012
*
arXiv
*
pre-print

We initiate

arXiv:1112.0230v2
fatcat:j72ftxce7fgf7gcwgbeexyf7he
*a*study*of**the*rings*of**invariants**of**modular*representations*of*elementary abelian p-*groups*. ... With*a*few notable exceptions,*the**modular*representation theory*of*an elementary abelian p-*group*is wild. However, for*a*given dimension, it is possible to parameterise*the*representations. ... For*a*faithful*modular*three dimensional representation*of*(Z/p) r ,*the*ring*of**invariants*is*a*complete intersection with embedding dimension less than or equal to ⌈r/2⌉ + 3. ...##
###
Braid group representations from twisted tensor products of algebras
[article]

2019
*
arXiv
*
pre-print

Our results hint at

arXiv:1906.08153v1
fatcat:mfedkiozirc5tdluuqkudsihey
*a*relationship between*the*braidings on*the*G-gaugings*of**a*pointed*modular*category C(*A*,Q) and that*of*C(*A*,Q) itself. ... We unify and generalize several approaches to constructing braid*group*representations from finite*groups*, using iterated twisted tensor products. ... )× Q(q)[x g : g ∈ G], using generators and relations, with*the*x g being*commuting*variables. (2) Define r i = g∈G x g g i for i = 1, 2, and use*non*-*commutative**Gröbner**bases*to write r 1 r 2 r 1 − r ...##
###
Braid Group Representations from Twisted Tensor Products of Algebras

2020
*
Peking Mathematical Journal
*

Our results suggest

doi:10.1007/s42543-020-00023-5
fatcat:7qzvfa5kaffvnmhhct6joslqiu
*a*relationship between*the*braiding on*the*G-gaugings*of**a*pointed*modular*category C(*A*, Q) and that*of*C(*A*, Q) itself. ... We provide some general characterizations and classification*of*these representations, focusing on*the*size*of*their images, which are typically finite*groups*. ...*Gröbner**bases*to write r 1 r 2 r 1 − r 2 r 1 r 2 in its normal form, i.e., as*a*polynomial in*the*g (i 1 ) 1 g (i 2 ) 2 with coefficients in ℚ(q)[x g ∶ g ∈ G]. (3) Compute*a**commutative**Gröbner*basis ...##
###
Rational invariants of finite abelian groups

2015
*
ACM Communications in Computer Algebra
*

We investigate

doi:10.1145/2768577.2768641
fatcat:fnqwoj5mrjea3clxypdipecc6y
*the*computation and applications*of*rational*invariants**of**the*linear action*of**a*finite abelian*group*in*the**non*-*modular**case*. ... We make use*of*linear algebra to compute*a*minimal generating set*of**invariants*and*the*substitution to rewrite any*invariant*in terms*of*this generating set. ... Acknowledgement: Part*of*this research was conducted when both authors were hosted by*the*Institute*of*Mathematical Sciences in*the*National University*of*Singapore during*the*amazing program Inverse Moment ...##
###
Fast Computation of Secondary Invariants
[article]

2007
*
arXiv
*
pre-print

*A*very classical subject in

*Commutative*Algebra is

*the*

*Invariant*Theory

*of*finite

*groups*. In our work on 3-dimensional topology (S. King,

*Ideal*Turaev-Viro

*invariants*. To appear in Top. ...

*The*implementation

*of*our algorithm in Singular is for

*the*

*non*-

*modular*

*case*; however,

*the*key theorem

*of*our algorithm holds in

*the*

*modular*

*case*as well and might be useful also there. ... I'm grateful to Gregor Kemper for providing me with

*the*data

*of*Example (9) . I owe thanks to Gregor Kemper and Nicolas Thiéry for their comments on this manuscript. ...

##
###
Around the numeric–symbolic computation of differential Galois groups

2007
*
Journal of symbolic computation
*

In particular, we present

doi:10.1016/j.jsc.2006.03.007
fatcat:hwkpzai7trbhjnwzjh77wshciu
*a**non*-heuristic algorithm for*the*factorization*of*linear differential operators. ... It can be shown that*the*differential Galois*group**of*L is generated (as*a*closed algebraic*group*) by*a*finite number*of*monodromy matrices, Stokes matrices and matrices in local exponential*groups*. ... Acknowledgments*The*author would like to thank J.-Y. Hée, W. de Graaf, F. Zara, J.Écalle and*the*referees for several helpful comments and references. ...
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