共查询到20条相似文献,搜索用时 15 毫秒
1.
Françcois Jaeger 《Journal of Algebraic Combinatorics》2000,11(3):241-268
We study the four-weight spin models (W1, W2, W3, W4) introduced by Eiichi and Etsuko Bannai (Pacific J. of Math, to appear). We start with the observation, based on the concept of special link diagram, that two such spin models yield the same link invariant whenever they have the same pair (W1, W3), or the same pair (W2, W4). As a consequence, we show that the link invariant associated with a four-weight spin model is not sensitive to the full reversal of orientation of a link. We also show in a similar way that such a link invariant is invariant under mutation of links.Next, we give an algebraic characterization of the transformations of four-weight spin models which preserve W1, W3 or preserve W2, W4. Such gauge transformations correspond to multiplication of W2, W4 by permutation matrices representing certain symmetries of the spin model, and to conjugation of W1, W3 by diagonal matrices. We show for instance that up to gauge transformations, we can assume that W1, W3 are symmetric.Finally we apply these results to two-weight spin models obtained as solutions of the modular invariance equation for a given Bose-Mesner algebra B and a given duality of B. We show that the set of such spin models is invariant under certain gauge transformations associated with the permutation matrices in B. In the case where B is the Bose-Mesner algebra of some Abelian group association scheme, we also show that any two such spin models (which generalize those introduced by Eiichi and Etsuko Bannai in J. Alg. Combin. 3 (1994), 243–259) are related by a gauge transformation. As a consequence, the link invariant associated with such a spin model depends only trivially on the link orientation. 相似文献
2.
Zdravka Boz
ikov 《组合设计杂志》1998,6(4):231-233
According to Mathon and Rosa [The CRC handbook of combinatorial designs, CRC Press, 1996] there is only one known symmetric design with parameters (69, 17, 4). This known design is given in Beth, Jungnickel, and Lenz [Design theory, B. I. Mannheim, 1985]; the Frobenius group F39 of order 39 acts on this design, where Z13 has exactly 4 fixed points and Z3 has exactly 9 fixed points. The purpose of this article is to investigate the converse of this fact with the hope of obtaining in this way at least one more design with these parameters. In fact we obtain exactly one new such design. In this article we have classified all such designs invariant under F39. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 231–233, 1998 相似文献
3.
We characterize the SDP designs that give rise to four-weight spin models with two values. We prove that the only such designs are the symplectic SDP designs. The proof involves analysis of the cardinalities of intersections of four blocks.AMS classification:05B20, 05E30 相似文献
4.
5.
K. Nomura 《Journal of Algebraic Combinatorics》1995,4(2):173-182
A spin model is one of the statistical mechanical models which were introduced by V.F.R. Jones to construct invariants of links. In this paper, we give a new construction of spin models of size 4n from a given spin model of size n. The process is similar to taking tensor product with a spin model of size four, but we add some sign exchange. This construction also gives symmetric four-weight spin models of the type introduced by E. Bannai and E. Bannai. 相似文献
6.
A biplane is a 2‐(k(k ? 1)/2 + 1,k,2) symmetric design. Only sixteen nontrivial biplanes are known: there are exactly nine biplanes with k < 11, at least five biplanes with k = 11, and at least two biplanes with k = 13. It is here shown by exhaustive computer search that the list of five known biplanes with k = 11 is complete. This result further implies that there exists no 3‐(57, 12, 2) design, no 11211 symmetric configuration, and no (324, 57, 0, 12) strongly regular graph. The five biplanes have 16 residual designs, which by the Hall–Connor theorem constitute a complete classification of the 2‐(45, 9, 2) designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 117–127, 2008 相似文献
7.
We extend the notion of unital as well as unitary polarity from finite projective planes to arbitrary symmetric designs. The existence of unitals in several families of symmetric designs has been proved. It is shown that if a unital in a point-hyperplane design PG
d-1(d,q) exists, then d = 2 or 3; in particular, unitals and ovoids are equivalent in case d = 3. Moreover, unitals have been found in two designs having the same parameters as the PG
4(5,2), although the latter does not have a unital. It had been not known whether or not a nonclassical design exists, which has a unitary polarity. Fortunately, we have discovered a unitary polarity in a symmetric 2-(45,12,3) design. To a certain extent this example seems to be exceptional for designs with these parameters. 相似文献
8.
R. Bruce Richter 《Journal of Graph Theory》1994,18(7):751-755
In this article, we develop a theory of walks traversing every edge exactly twice. Rosenstiehl and Read proved that if G is a graph with no set of edges that is simultaneously a cycle and a cocycle, then G is planar if and only if there is a closed walk W in G traversing every edge exactly twice such that certain sets of edges derived from W are all cocycles. One consequence of the current work is a simple proof of the Rosenstiehl-Read theorem. Another is an unusual method for determining the rank (over the integers modulo 2) of a symmetric matrix obtained from a circle graph. 相似文献
9.
Etsuko Bannai 《Graphs and Combinatorics》2001,17(4):589-598
It is known that for each matrix W
i
and it's transpose
t
W
i
in any four-weight spin model (X, W
1, W
2, W
3, W
4; D), there is attached the Bose-Mesner algebra of an association scheme, which we call Nomura algebra. They are denoted by N(W
i
) and N(
t
W
i
) = N′(W
i
) respectively. H. Guo and T. Huang showed that some of them coincide with a self-dual Bose-Mesner algebra, that is, N(W
1) = N′(W
1) = N(W
3) = N′(W
3) holds. In this paper we show that all of them coincide, that is, N(W
i
), N′(W
i
), i=1, 2, 3, 4, are the same self-dual Bose-Mesner algebra.
Received: June 17, 1999 Final version received: Januray 17, 2000 相似文献
10.
A spin model is a triple (X, W
+, W
–), where W
+ and W
– are complex matrices with rows and columns indexed by X which satisfy certain equations (these equations allow the construction of a link invariant from(X, W
+, W
–) ). We show that these equations imply the existence of a certain isomorphism between two algebras
and
associated with (X, W
+, W
–) . When
is the Bose-Mesner algebra of some association scheme, and is a duality of
. These results had already been obtained in [15] when W
+, W
– are symmetric, and in [5] in the general case, but the present proof is simpler and directly leads to a clear reformulation of the modular invariance property for self-dual association schemes. This reformulation establishes a correspondence between the modular invariance property and the existence of spin models at the algebraic level. Moreover, for Abelian group schemes, spin models at the algebraic level and actual spin models coincide. We solve explicitly the modular invariance equations in this case, obtaining generalizations of the spin models of Bannai and Bannai [3]. We show that these spin models can be identified with those constructed by Kac and Wakimoto [20] using even rational lattices. Finally we give some examples of spin models at the algebraic level which are not actual spin models. 相似文献
11.
Kazumasa Nomura 《Graphs and Combinatorics》1999,15(1):79-92
. A type II matrix is an n×n matrix W with non-zero entries W
i,j which satisfies , i, j=1, …, n. Two type II matrices W, W′ are said to be equivalent if W′=P
1Δ1
WΔ2
P
2 holds for some permutation matrices P
1, P
2 and for some non-singular diagonal matrices Δ1, Δ2. In the present paper, it is shown that there are up to equivalence exactly three type II matrices in M
5(C).
Received: August 15, 1996 Revised: May 16, 1997 相似文献
12.
Symmetric nets are affine resolvable designs whose duals are also affine. It is shown that. up to isomorphism, there are exactly four symmetric (3, 3)-nets (v=b=27,k=9), and exactly two inequivalent 9×9 generalized Hadamard matrices over the group of order 3. The symmetric (3, 3)-nets are found as subnets of affine resolvable 2-(27, 9, 4) designs. Ten of the 68 non-isomorphic affine resolvable 2-(27, 9, 4) designs are not extensions of symmetric (3, 3)-subnets, providing the first examples of affine 2-(q3, q2, q2–1/q–1) designs without symmetric (q, q)-subnets. 相似文献
13.
A symmetric design with parameters v = q
2(q + 2), k = q(q + 1), λ = q, q ≥ 2, is called a quasi-affine design if its point set can be partitioned into q + 2 subsets P
0, P
1,..., P
q
, P
q+1 such that the induced structure in every point neighborhood is an affine plane of order q (repeated q times). A quasi-affine design with q ≥ 3 determines its point neighborhoods uniquely and dual of such a design is also a quasi-affine design. These structural
properties pave way for definition of a strongly quasi-affine design and it is also shown that associated with every quasi-affine
design is a unique strongly quasi-affine design from which the given quasi-affine design is obtained by certain unique cutting
and pasting operation. This investigation also enables us to associate a unique 2-regular graph with q + 2 vertices and in turn, a unique colored partition of the integer q + 2. These combinatorial consequences are finally used to obtain an exponential lower bound on the number of non-isomorphic
solutions of such symmetric designs improving the earlier lower bound of 2.
Work of Sanjeevani Gharge is supported by Faculty Improvement Programme of U.G.C., India. 相似文献
14.
A spin model (for link invariants) is a square matrix W which satisfies certain axioms. For a spin model W, it is known that W T W ?1 is a permutation matrix, and its order is called the index of W. Jaeger and Nomura found spin models of index?2, by modifying the construction of symmetric spin models from Hadamard matrices. The aim of this paper is to give a construction of spin models of an arbitrary even index from any Hadamard matrix. In particular, we show that our spin models of indices a power of 2 are new. 相似文献
15.
An (α,n)-arc in a 2-design is a set ofn points of the design such that any block intersects it in at most α points. For such an arc,n is bounded by 1+(r(α−1)/λ), with equality if and only if every block meets the arc in either 0 or α points. An (α,n) arc with equality in above is said to be maximal.
A maximal block arc can be dually defined. This generalizes the notion of an oval (α=2) in a symmetric design due to Asmus
and van Lint. The aim of this paper is to study the infinite family of possibly extendable symmetric designs other than the
Hadamard design family and their related designs using maximal arcs. It is shown that the extendability corresponds to the
existence of a proper family of maximal arcs. A natural duality between point and block arcs is established, which among other
things implies a result of Cameron and van Lint that extendability of a given design in this family is equivalent to extendability
of its dual. Similar results are proved for other related designs. 相似文献
16.
Hannah Alpert 《Archiv der Mathematik》2011,96(4):311-320
Let A be an infinite set that generates a group G. The sphere S
A
(r) is the set of elements of G for which the word length with respect to A is exactly r. We say G admits all finite transitions if for every r ≥ 2 and every finite symmetric subset W ì G\{e}{W \subset G{\setminus}\{e\}}, there exists an A with S
A
(r) = W. In this paper we determine which countable abelian groups admit all finite transitions. We also show that
\mathbbRn{\mathbb{R}^n} and the finitary symmetric group on
\mathbbN{\mathbb{N}} admit all finite transitions. 相似文献
17.
In this paper, we investigate the existence of large sets of symmetric partitioned incomplete latin squares of type gu (LSSPILSs) which can be viewed as a generalization of the well‐known golf designs. Constructions for LSSPILSs are presented from some other large sets, such as golf designs, large sets of group divisible designs, and large sets of Room frames. We prove that there exists an LSSPILS(gu) if and only if u ≥ 3, g(u ? 1) ≡ 0 (mod 2), and (g, u) ≠ (1, 5). 相似文献
18.
A Menon design of order h2 is a symmetric (4h2,2h2‐h,h2‐h)‐design. Quasi‐residual and quasi‐derived designs of a Menon design have parameters 2‐(2h2 + h,h2,h2‐h) and 2‐(2h2‐h,h2‐h,h2‐h‐1), respectively. In this article, regular Hadamard matrices are used to construct non‐embeddable quasi‐residual and quasi‐derived Menon designs. As applications, we construct the first two new infinite families of non‐embeddable quasi‐residual and quasi‐derived Menon designs. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 53–62, 2009 相似文献
19.
We propose a technique for constructing two infinite families of non‐embeddable quasi‐residual designs as soon as one such design satisfying certain conditions exists. The main tools are generalized Hadamard matrices and balanced generalized weighing matrices. Starting with a specific non‐embeddable quasi‐residual 2‐(27,9,4) design, we construct for every positive integer m a non‐embeddable 2‐(3m,3m?1,(3m?1?1)/2)‐design, and, if rm=(3m?1)/2 is a prime power, we construct for every positive integer n a non‐embeddable design. For each design in these families, a symmetric design with the corresponding parameters is known to exist. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 160–172, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.900 相似文献
20.
Jo Ann W. Staples 《Journal of Graph Theory》1979,3(2):197-204
A set of points in a graph is independent if no two points in the set are adjacent. A graph is well covered if every maximal independent set is a maximum independent set or, equivalently, if every independent set is contained in a maximum independent set. The well-covered graphs are classified by the Wn property: For a positive integer n, a graph G belongs to class Wn if ≥ n and any n disjoint independent sets are contained in n disjoint maximum independent sets. Constructions are presented that show how to build infinite families of Wn graphs containing arbitrarily large independent sets. A characterization of Wn graphs in terms of well-covered subgraphs is given, as well as bounds for the size of a maximum independent set and the minimum and maximum degrees of points in Wn graphs. 相似文献