Graph Theory

Geometry of semilinear embeddings : relations to graphs and by Mark Pankov

By Mark Pankov

This quantity covers semilinear embeddings of vector areas over department jewelry and the linked mappings of Grassmannians. not like classical books, we think of a extra normal category of semilinear mappings and express that this type is critical. a wide element of the cloth should be formulated by way of graph concept, that's, Grassmann graphs, graph embeddings, and isometric embeddings. moreover, a few kin to linear codes may be defined. Graduate scholars and researchers will locate this quantity to be self-contained with many examples.

Readership: Graduate scholars and researchers attracted to the sphere of semilinear embeddings

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Extra resources for Geometry of semilinear embeddings : relations to graphs and codes

Example text

7. The following assertions are fulfilled: (1) the mapping (l)k is injective for every k ∈ {1, . . , m − 1}, (2) if m < n and (l)m is injective then l is a semilinear (m + 1)-embedding. Proof. (1). If (l)k is non-injective for a certain k ∈ {1, . . , m − 1} then there exist distinct k-dimensional subspaces S, U ⊂ V such that l(S) and l(U ) span the same k-dimensional subspace of V ′ . 2, this k-dimensional subspace coincides with l(S + U ) . Hence it contains the image of every (k + 1)-dimensional subspace T ⊂ S + U (the dimension of S + U is not less than k + 1, since S and U are distinct).

If p ∈ s then f (s) coincides with l. e. the restriction of f to s is not injective or constant. The latter means that f cannot be induced by a semilinear transformation of the associated vector space. 2 implies the existence of a semilinear injection l : V → V ′ such that f = (l)1 . Since f transfers any triple of non-collinear points to a triple of non-collinear points, l is a semilinear 3-embedding. 8 and get the classical version of the Fundamental Theorem of Projective Geometry. 2. Every collineation of ΠV to ΠV ′ is induced by a semilinear isomorphism of V to V ′ .

M − 1} there exists U ∈ Gk (E m ) such that the dimension of the subspace U ∩ F m in the vector space F m is less than k. page 22 March 19, 2015 14:43 BC: 9465 - Geometry of Semilinear Embeddings Semilinear mappings main 23 Proof. 1, the mapping of Gk (F m ) to Gk (E m ) induced by the identity embedding of F m in E m is non-surjective. If a ∈ R′ is non-zero then (al)k coincides with (l)k for all admissible k. 9. If s : V → V ′ is a semilinear m-embedding such that (s)k coincides with (l)k for a certain k ∈ {1, .

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