303 research outputs found
A seminormal form for partition algebras
Using a new presentation for partition algebras (J. Algebraic Combin.
37(3):401-454, 2013), we derive explicit combinatorial formulae for the
seminormal representations of the partition algebras. These results generalise
to the partition algebras the classical formulae given by Young for the
symmetric group.Comment: Published version. 51 pages, includes figures and table
Seminormal forms and Gram determinants for cellular algebras
This paper develops an abstract framework for constructing ``seminormal
forms'' for cellular algebras. That is, given a cellular R-algebra A which is
equipped with a family of JM-elements we give a general technique for
constructing orthogonal bases for A, and for all of its irreducible
representations, when the JM-elements separate A. The seminormal forms for A
are defined over the field of fractions of R. Significantly, we show that the
Gram determinant of each irreducible A-module is equal to a product of certain
structure constants coming from the seminormal basis of A. In the non-separated
case we use our seminormal forms to give an explicit basis for a block
decomposition of A.
The appendix, by Marcos Soriano, gives a general construction of a complete
set of orthogonal idempotents for an algera starting from a set of elements
which act on the algebra in an upper triangular fashion. The appendix shows
that constructions with "Jucys-Murphy elements"depend, ultimately, on the
Cayley-Hamilton theorem.Comment: Final version. To appear J. Reine Angew. Math. Appendix by Marcos
Sorian
Degenerate two-boundary centralizer algebras
Diagram algebras (e.g. graded braid groups, Hecke algebras, Brauer algebras)
arise as tensor power centralizer algebras, algebras of commuting operators for
a Lie algebra action on a tensor space. This work explores centralizers of the
action of a complex reductive Lie algebra on tensor space of the
form . We define the degenerate two-boundary
braid algebra and show that centralizer algebras contain
quotients of this algebra in a general setting. As an example, we study in
detail the combinatorics of special cases corresponding to Lie algebras
and and modules and indexed by
rectangular partitions. For this setting, we define the degenerate extended
two-boundary Hecke algebra as a quotient of
, and show that a quotient of is
isomorphic to a large subalgebra of the centralizer. We further study the
representation theory of to find that the
seminormal representations are indexed by a known family of partitions. The
bases for the resulting modules are given by paths in a lattice of partitions,
and the action of is given by combinatorial
formulas.Comment: 45 pages, to appear in Pacific Journal of Mathematic
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