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Gröbner bases of ideals invariant under a commutative group

Jean-Charles Faugere, Jules Svartz
2013 Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13  
We propose efficient algorithms to compute 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.  ... 
doi:10.1145/2465506.2465944 dblp:conf/issac/FaugereS13 fatcat:mej5g5bxmzhwxoj67vqta5zgeu

Calculating Invariant Rings of Finite Groups over Arbitrary Fields

GREGOR KEMPER
1996 Journal of symbolic computation  
Up to now, such algorithms have been available only for 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.  ... 
doi:10.1006/jsco.1996.0017 fatcat:4qx3ifrlkned3frja4mljenmby

The isomorphism problem for graded algebras and its application to mod-p cohomology rings of small p-groups [article]

Bettina Eick, Simon King
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  ... 
arXiv:1503.04666v1 fatcat:owfi4grsq5ajxneztc2fhkflwm

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

Arpit Dua and Pratyush Sarkar and Dominic J. Williamson and Meng Cheng
2019 arXiv   pre-print
We calculate bifurcating entanglement-renormalization 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).  ... 
arXiv:1909.12304v1 fatcat:unz7muuyubamvepz3yfuewsose

Rings of invariants for modular representations of elementary abelian p-groups

H. E. A. Campbell, R. J. Shank, D. L. Wehlau
2013 Transformation groups  
We initiate 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.  ... 
doi:10.1007/s00031-013-9207-z fatcat:rta7t5jrzzcmpfdzs33fdothra

Bifurcating entanglement-renormalization group flows of fracton stabilizer models

Arpit Dua, Pratyush Sarkar, Dominic J. Williamson, Meng Cheng
2020 Physical Review Research  
We calculate bifurcating entanglement-renormalization 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.  ... 
doi:10.1103/physrevresearch.2.033021 fatcat:tnawxsfg7nho3ho6rgsw4j35zy

Computing Modular Invariants of p-groups

R James Shank, David L Wehlau
2002 Journal of symbolic computation  
We show that there exists 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.  ... 
doi:10.1006/jsco.2002.0558 fatcat:zcrkmcs7mzee7huedtjnnfwpmm

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

R. JAMES SHANK, DAVID L. WEHLAU
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.  ... 
doi:10.1112/s0024609302001054 fatcat:ltinftey5jaq7jodbjyfsvntd4

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

H. E. A. Campbell, R. J. Shank, D. L. Wehlau
2012 arXiv   pre-print
We initiate 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.  ... 
arXiv:1112.0230v2 fatcat:j72ftxce7fgf7gcwgbeexyf7he

Braid group representations from twisted tensor products of algebras [article]

Paul Gustafson, Andrew Kimball, Eric C. Rowell, Qing Zhang
2019 arXiv   pre-print
Our results hint at 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  ... 
arXiv:1906.08153v1 fatcat:mfedkiozirc5tdluuqkudsihey

Braid Group Representations from Twisted Tensor Products of Algebras

Paul Gustafson, Andrew Kimball, Eric C. Rowell, Qing Zhang
2020 Peking Mathematical Journal  
Our results suggest 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  ... 
doi:10.1007/s42543-020-00023-5 fatcat:7qzvfa5kaffvnmhhct6joslqiu

Rational invariants of finite abelian groups

George Labahn
2015 ACM Communications in Computer Algebra  
We investigate 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  ... 
doi:10.1145/2768577.2768641 fatcat:fnqwoj5mrjea3clxypdipecc6y

Fast Computation of Secondary Invariants [article]

Simon A. King
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.  ... 
arXiv:math/0701270v4 fatcat:wwoo5if6vvat3lci6atczg4vyu

Around the numeric–symbolic computation of differential Galois groups

Joris van der Hoeven
2007 Journal of symbolic computation  
In particular, we present 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.  ... 
doi:10.1016/j.jsc.2006.03.007 fatcat:hwkpzai7trbhjnwzjh77wshciu
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