Geodesic
Degrees of Irreducible Characters and Dimensions of Hadamard Algebras
Matematičeskie zametki, Tome 98 (2015) no. 2, pp. 230-236

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The notion of Hadamard decomposition of a semisimple associative finite-dimensional complex algebra generalizes the notion of classical Hadamard matrix corresponding to the case of commutative algebras. The algebras admitting a Hadamard decomposition are referred to as Hadamard algebras. We study the conjecture claiming that, if a Hadamard algebra is not simple and has an irreducible character of degree $m\ge 2$, then the dimension of the algebra is not less than $2m^2$. The validity of this conjecture is confirmed for the first two values $m=2$ and $m=4$ (here $m$ must be even). Moreover, we prove a result (which is weaker than the conjecture) in which $2m^2$ is replaced by $m^2+2m$.
Mots-clés : Hadamard decomposition, Hadamard algebra, Hadamard matrix
Keywords: irreducible character. 
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     author = {D. N. Ivanov},
     title = {Degrees of {Irreducible} {Characters} and {Dimensions} of {Hadamard} {Algebras}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {230--236},
     publisher = {mathdoc},
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     year = {2015},
     language = {ru},
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D. N. Ivanov. Degrees of Irreducible Characters and Dimensions of Hadamard Algebras. Matematičeskie zametki, Tome 98 (2015) no. 2, pp. 230-236. http://geodesic.mathdoc.fr/item/MZM_2015_98_2_a7/