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Homomorphisms from the unitary group to the general linear group over complex number field and applications

Chong-Guang Cao, Xian Zhang (2002)

Archivum Mathematicum

Let M n be the multiplicative semigroup of all n × n complex matrices, and let U n and G L n be the n –degree unitary group and general linear group over complex number field, respectively. We characterize group homomorphisms from U n to G L m when n > m 1 or n = m 3 , and thereby determine multiplicative homomorphisms from U n to M m when n > m 1 or n = m 3 . This generalize Hochwald’s result in [Lin. Alg. Appl.  212/213:339-351(1994)]: if f : U n M n is a spectrum–preserving multiplicative homomorphism, then there exists a matrix R in G L n such that f ( A ) = R A R for...

Horocyclic products of trees

Laurent Bartholdi, Markus Neuhauser, Wolfgang Woess (2008)

Journal of the European Mathematical Society

Let T 1 , , T d be homogeneous trees with degrees q 1 + 1 , , q d + 1 3 , respectively. For each tree, let 𝔥 : T j be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of T 1 , , T d is the graph 𝖣𝖫 ( q 1 , , q d ) consisting of all d -tuples x 1 x d T 1 × × T d with 𝔥 ( x 1 ) + + 𝔥 ( x d ) = 0 , equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If d = 2 and q 1 = q 2 = q then we obtain a Cayley graph of the...

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