Growth Functions of Finitely Generated Algebras

Advances in Mathematics, 2008

Let A be an algebra over a field F of characteristic zero and let c n (A), n = 1, 2,. .. , be its sequence of codimensions. We prove that if c n (A) is exponentially bounded, its exponential growth can be any real number > 1. This is achieved by constructing, for any real number α > 1, an F-algebra A α such that lim n→∞ n √ c n (A α) exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.

Journal of Algebra, 1999

2006

Proceedings of the American Mathematical Society, 2000

Let A be an associative algebras over a field of characteristic zero. We prove that the codimensions of A are polynomially bounded if and only if any finite dimensional algebra B with Id(A) = Id(B) has an explicit decomposition into suitable subalgebras; we also give a decomposition of the n-th cocharacter of A into suitable Sn-characters. We give similar characterizations of finite dimensional algebras with involution whose *-codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group.

Advances in Applied Mathematics, 2006

Let F be a field of characteristic zero and let A be an F-algebra. The polynomial identities satisfied by A can be measured through the asymptotic behavior of the sequence of codimensions and the sequence of colengths of A. For finite dimensional algebras we show that the colength sequence of A is polynomially bounded and the codimension sequence cannot have intermediate growth. We then prove that for general nonassociative algebras intermediate growth of the codimensions is allowed. In fact, for any real number 0 < β < 1, we construct an algebra A whose sequence of codimensions grows like n n β .

Advances in Applied Mathematics, 2011

Let A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded. Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One of the main tools of independent interest is the construction in the free non-associative algebra of multialternating polynomials satisfying special properties. addresses: [email protected] (A. Giambruno), [email protected] (I. Shestakov), [email protected] (M. Zaicev).

2008

The profile of a relational structure R is the function phi which counts for every integer n the number, possibly infinite, phi(n) of substructures of R induced on the n-element subsets, isomorphic substructures being identified. If phi takes only finite values, this is the Hilbert function of a graded algebra associated with R, the age algebra, introduced by P. J. Cameron. In this paper, we give a closer look at this association, particularly when the relational structure R admits a finite monomorphic decomposition. This setting still encompass well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. We prove that phi is eventually a quasi-polynomial, this supporting the conjecture that, under mild assumptions on R, phi is eventually a quasi-polynomial whenever it is bounded by some polynomial. We also characterize when the age algebra is finitely generated.

2006

The profile of a relational structure R is the function phi_R which counts for every integer n the number, possibly infinite, phi_R(n) of substructures of R induced on the n-element subsets, isomorphic substructures being identified. Several graded algebras can be associated with R in such a way that the profile of R is simply the Hilbert function. An example of such graded algebra is the age algebra introduced by P.~J.~Cameron. In this paper, we give a closer look at this association, particularly when the relational structure R decomposes into finitely many monomorphic components. In this case, several well-studied graded commutative algebras (e.g. the invariant ring of a finite permutation group, the ring of quasi-symmetric polynomials) are isomorphic to some age algebras. Also, phi_R is a quasi-polynomial, this supporting the conjecture that, with mild assumptions on R, phi_R is a quasi-polynomial when it is bounded by some polynomial.

Transactions of the American Mathematical Society, 1985

Let G G be an associative monomial k {\mathbf {k}} -algebra. If G G is assumed to be finitely presented, then either G G contains a free subalgebra on two monomials or else G G has polynomial growth. If instead G G is assumed to have finite global dimension, then either G G contains a free subalgebra or else G G has a finite presentation and polynomial growth. Also, a graded Hopf algebra with generators in degree one and relations in degree two contains a free Hopf subalgebra if the number of relations is small enough.

The electronic journal of combinatorics

The profile of a relational structure R is the function ϕ R which counts for every integer n the number ϕ R (n), possibly infinite, of substructures of R induced on the n-element subsets, isomorphic substructures being identified. If ϕ R takes only finite values, this is the Hilbert function of a graded algebra associated with R, the age algebra KA(R), introduced by P. J. Cameron.