arXiv subjects

Sören Schwenker

Publications and source records attributed to Sören Schwenker.

3 recordsLinked to original sources

arXiv2019

A new algorithm for computing idempotents of R-trivial monoids

The authors of [Primitive orthogonal idempotents for R-trivial monoids, Journal of Algebra] provide an algorithm for finding a complete system of primitive orthogonal idempotents for CM, where M is any finite R-trivial monoid. Their method relies on a technical result stating that R-trivial monoids are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an R-trivial monoid may be realised by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where L-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We also show that our algorithm only has to be performed for ZM, after which a complete system of primitive orthogonal idempotents may be obtained for RM, where R is any ring with a known complete system of primitive orthogonal idempotents. In particular, the algorithm works if R is any field.

math.RT↗

arXiv2018

Generic steady state bifurcations in monoid equivariant dynamics with applications in homogeneous coupled cell systems

We prove that steady state bifurcations in finite-dimensional dynamical systems that are symmetric with respect to a monoid representation generically occur along an absolutely indecomposable subrepresentation. This is stated as a conjecture in B. Rink and J. Sanders, "Coupled cell networks and their hidden symmetries", SIAM J. Math. Anal., 46 (2014). It is a generalization of the well-known fact that generic steady state bifurcations in equivariant dynamical systems occur along an absolutely irreducible subrepresentation if the symmetries form a group - finite or compact Lie. Our generalization also includes non-compact symmetry groups. The result has applications in bifurcation theory of homogeneous coupled cell networks as they can be embedded (under mild additional assumptions) into monoid equivariant systems.

math.DS↗

arXiv2013

Equivariant Bifurcation and Absolute Irreducibility in $\R^8$ A contribution to the Ize conjecture

M. Field [5] refers to an unpublished work by J. Ize for a result that loss of stability through an absolutely irreducible representation of a compact Lie group leads to bifurcation of steady states. The main ingredient of the proof is the hypotheses, that for an absolutely irreducible representation of a compact Lie group there exists a closed subgroup whose fixed point space is odd dimensional. Then, using Brouwer degree, one gets the result. We refer to the hypotheses that for an absolutely irreducible representation of a compact Lie group there exists at least one subgroup with an odd dimensional fixed point space as the algebraic Ize conjecture (AIC). Lauterbach and Matthews [10] have shown that the (AIC) is in general not true. In fact they have constructed three infinite families of finite subgroups of SO{4} which act absolutely irreducibly on $\R^4$ and for each of them any isotropy subgroup has an even dimensional fixed point space. Moreover in [10] it is shown that in spite of this failure of the (AIC) the original conjecture is true at least for groups in two of these three families. In this paper we show a similar bifurcation result for the third family defined in [10]. We go on and construct a family of groups acting absolutely irreducibly on $\R^8$ which have only even dimensional fixed point spaces. Then we discuss the steady state bifurcations in this case. We end this paper with a discussion on how to extend the results in [10] to larger sets of groups which act on $\R^{4}$.

math.DS↗