7.6.3 Letterplace correspondence
Our work as well as the name letteplace has been inspired by the work of Rota.
Already Feynman and Rota encoded
the monomials (words) of the free algebra
via the doubleindexed letterplace
(that is encoding the letter (= variable) and its place in the word) monomials
, where
and is the monoid of natural numbers, starting with 0 which cannot be used as a place.
Note, that the letterplace algebra
is an infinitely generated commutative polynomial
algebra.
Since
,...,
is not Noetherian, it is common to perform the computations with modules up to a given degree.
In that case the truncated letterplace algebra is finitely generated commutative ring.
In [LL09] a natural shifting on letterplace polynomials was introduced and used.
Indeed, there is 1to1 correspondence between graded twosided ideals
of a free algebra and socalled letterplace ideals in the letterplace algebra, see [LL09] for details.
All the computations take place in the letterplace algebra.
A letterplace monomial of length
is a monomial of a letterplace algebra, such that its
places are exactly 1,2,...,
. In particular, such monomials are multilinear with respect to places. A letterplace polynomial is an element of the
vector space, spanned by letterplace monomials. A letterplace ideal is generated by letterplace polynomials subject to two kind of operations:
the
algebra operations of the letterplace algebra and simultaneous shifting of places by any natural number .
