The k-orbit reconstruction and the orbit algebra

Autor: V. B. Mnukhin
Rok vydání: 1992
Předmět:
Zdroj: Acta Applicandae Mathematicae. 29:83-117
ISSN: 1572-9036
0167-8019
DOI: 10.1007/bf00053380
Popis: Let (G, W) be a permutation group on a finite set W = {w 1,..., w n}. We consider the natural action of G on the set of all subsets of W. Let h 0, h 1,..., h N be the orbits of this action. For each i, 1 ≤ i ≤ N, there exists k, 1 ≤ k ≤ n, such that h i is a set of k-element subsets of W. In this case h i is called a symmetrized k-orbit of the group (G, W) or simply a k-orbit. With a k-orbit h i we associate a multiset H(h i ) = % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeoaaa!3690!\[\langle \]h i (1), h i (2),..., h i (k)% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOkJepaaa!36A1!\[\rangle \] of its (k − 1)-suborbits. Orbits h i and h j are called equivalent if H(h i ) = H(h j ). An orbit is reconstructible if it is equivalent to itself only. The paper concerns the k-orbit reconstruction problem and its connections with different problems in combinatorics. The technique developed is based on the notion of orbit and co-orbit algebras associated with a given permutation group (G, W).
Databáze: OpenAIRE
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