The module structure of integral designs
TL;DR: The existence problem for t-designs with prescribed parameters is solved by allowing positive and negative integral multiplicities for the blocks.
About: This article is published in Journal of Combinatorial Theory, Series A.The article was published on 1973-07-01 and is currently open access. It has received 104 citations till now. The article focuses on the topics: Structure (category theory).
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01 Jan 1994
TL;DR: In this article, a formal theory of Tannaka duality inspired by Ross Street's "formal theory of monads" is presented. But the formal theory is restricted to the case where the representations of the representations are over an arbitrary commutative ring rather than a field.
Abstract: — A Tannakian category is an abelian tensor category equipped with a fiber functor and additional structures which ensure that it is equivalent to the category of representations of some affine groupoid scheme acting on the spectrum of a field extension. If we are working over an arbitrary commutative ring rather than a field, the categories of representations cease to be abelian. We provide a list of sufficient conditions which ensure that an additive tensor category is equivalent to the category of representations of an affine groupoid scheme acting on an affine scheme, or, more generally, to the category of representations of a Hopf algebroid in a symmetric monoidal category. In order to do this we develop a “formal theory of Tannaka duality” inspired by Ross Street’s “formal theory of monads.” We apply our results to certain categories of filtered modules which are used to study p-adic Galois representations. Résumé (La théorie formelle de dualité tannakienne). — Une catégorie tannakienne est une catégorie abélienne tensorielle munie d’un foncteur fibré et de structures additionnelles de manière à être équivalente à la catégorie des représentations d’un groupoïde affine agissant sur le spectre d’une extension de corps. Si l’on remplace les corps par des anneaux commutatifs, les catégories des représentations ne seront plus abéliennes. Nous donnons des conditions suffisantes pour qu’une catégorie additive tensorielle soit équivalente à la catégorie des représentations d’un schéma en groupoïdes affines, ou plus généralement, à la catégorie des représentations d’un algebroïde de Hopf dans une catégorie symmétrique monoïdale. Pour ce faire nous développons une « théorie formelle de dualité tannakienne » inspirée par la « théorie formelle des monades » de Ross Street. Nous appliquons nos résultats à certaines catégories des modules filtrés qui sont utilisées pour étudier les représentations galoisiennes p-adiques. © Astérisque 357, SMF 2013
463 citations
Cites background from "The module structure of integral de..."
...It is a common practice to look at a generalized notion of solution and Wilson [51] and Graver and Jurkat [22] asked for integral solutions....
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...(2) They proved: Theorem 3.3 (Wilson [50]; Graver and Jurkat [22])....
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...3 (Wilson [50]; Graver and Jurkat [22])....
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TL;DR: The bound (t+1)(k-t-1) represents the best possible strengthening of the original 1961 theorem of Erdös, Ko, and Rado which reaches the same conclusion under the hypothesisn≧t+(k−t) .
Abstract: This paper contains a proof of the following result: ifn≧(t+1)(k−t−1), then any family ofk-subsets of ann-set with the property that any two of the subsets meet in at leastt points contains at most\(\left( {\begin{array}{*{20}c} {n - t} \\ {k - t} \\ \end{array} } \right)\) subsets. (By a theorem of P. Frankl, this was known whent≧15.) The bound (t+1)(k-t-1) represents the best possible strengthening of the original 1961 theorem of Erdos, Ko, and Rado which reaches the same conclusion under the hypothesisn≧t+(k−t)\(\left( {\begin{array}{*{20}c} k \\ t \\ \end{array} } \right)^3 \). Our proof is linear algebraic in nature; it may be considered as an application of Delsarte’s linear programming bound, but somewhat lengthy calculations are required to reach the stated result. (A. Schrijver has previously noticed the relevance of these methods.) Our exposition is self-contained.
314 citations
Cites methods from "The module structure of integral de..."
...well-known (see, e.g., [ 4 ] or [9] ) and will be used again in Sections 4 and 5....
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Posted Content•
TL;DR: The existence conjecture for combinatorial designs has been proved in this article for simplicial designs, answering a question of Steiner from 1853 and showing that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition.
Abstract: We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition. As a further generalisation, we obtain the same conclusion only assuming an extendability property and the existence of a robust fractional clique decomposition.
278 citations
Cites background or methods or result from "The module structure of integral de..."
...In this section we generalise to the context of typical complexes the results of Graver and Jurkat and of Wilson on the module structure of integral designs....
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...We make some definitions and then state the main result of [13, 45]....
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...Next we reformulate the result of Graver and Jurkat, adding a boundedness condition that we need later....
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...The main result of this section is an analogue of the results of Graver and Jurkat [13] and Wilson [45] on integral decompositions in which we can also impose a boundedness requirement....
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...The following result of Graver and Jurkat [13] and Wilson [45] shows that the necessary divisibility conditions on J are sufficient for an integral decomposition Φ, i....
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TL;DR: The number of nonisomorphic designs on v points with given block size k > 2 and index λ tends to infinity as v increases (subject to the above congruences).
269 citations
Cites background from "The module structure of integral de..."
...(This observation is also essentially contained in [4])....
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TL;DR: A t-design T=(X, B), denoted by (@l; t, k, v), is a system B of subsets of size k from a v-set X, such that each t-subset of X is contained in exactly @l elements of B.
159 citations
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