7,067 research outputs found
Renormalization as a functor on bialgebras
The Hopf algebra of renormalization in quantum field theory is described at a
general level. The products of fields at a point are assumed to form a
bialgebra B and renormalization endows T(T(B)^+), the double tensor algebra of
B, with the structure of a noncommutative bialgebra. When the bialgebra B is
commutative, renormalization turns S(S(B)^+), the double symmetric algebra of
B, into a commutative bialgebra. The usual Hopf algebra of renormalization is
recovered when the elements of B are not renormalised, i.e. when Feynman
diagrams containing one single vertex are not renormalised. When B is the Hopf
algebra of a commutative group, a homomorphism is established between the
bialgebra S(S(B)^+) and the Faa di Bruno bialgebra of composition of series.
The relation with the Connes-Moscovici Hopf algebra of diffeomorphisms is
given. Finally, the bialgebra S(S(B)^+) is shown to give the same results as
the standard renormalisation procedure for the scalar field.Comment: 24 pages, no figure. Several changes in the connection with standard
renormalizatio
Vertices of Specht modules and blocks of the symmetric group
This paper studies the vertices, in the sense defined by J. A. Green, of
Specht modules for symmetric groups. The main theorem gives, for each
indecomposable non-projective Specht module, a large subgroup contained in one
of its vertices. A corollary of this theorem is a new way to determine the
defect groups of symmetric groups. We also use it to find the Green
correspondents of a particular family of simple Specht modules; as a corollary,
we get a new proof of the Brauer correspondence for blocks of the symmetric
group. The proof of the main theorem uses the Brauer homomorphism on modules,
as developed by M. Brou{\'e}, together with combinatorial arguments using Young
tableaux.Comment: 18 pages, 1 figur
Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras
Bell polynomials appear in several combinatorial constructions throughout
mathematics. Perhaps most naturally in the combinatorics of set partitions, but
also when studying compositions of diffeomorphisms on vector spaces and
manifolds, and in the study of cumulants and moments in probability theory. We
construct commutative and noncommutative Bell polynomials and explain how they
give rise to Fa\`a di Bruno Hopf algebras. We use the language of incidence
Hopf algebras, and along the way provide a new description of antipodes in
noncommutative incidence Hopf algebras, involving quasideterminants. We also
discuss M\"obius inversion in certain Hopf algebras built from Bell
polynomials.Comment: 37 pages, final version, to appear in IJA
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