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

**Read or Download Geometry of semilinear embeddings : relations to graphs and codes PDF**

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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, .