On the growth of the identities of algebras (original) (raw)

A characterization of algebras with polynomial growth of the codimensions

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.

An integral second fundamental theorem of invariant theory for partition algebras

arXiv: Representation Theory, 2018

We prove that the kernel of the action the group algebra of the Weyl group acting on tensor space (via restriction of the action from the general linear group) is a cell ideal with respect to the alternating Murphy basis. This provides an analogue of the second fundamental theory of invariant theory for the partition algebra over an arbitrary integral domain and proves that the centraliser algebras of the partition algebra are cellular. We also prove similar results for the half partition algebras.

COCHARACTERS OF POLYNOMIAL IDENTITIES OF UPPER TRIANGULAR MATRICES

Journal of Algebra and Its Applications, 2012

Let T (U k ) be the T-ideal of the polynomial identities of the algebra of k × k upper triangular matrices over a field of characteristic zero. We give an easy algorithm which calculates the generating function of the cocharacter sequence χn(U k ) = λ⊢n m λ (U k )χ λ of the T-ideal T (U k ). Applying this algorithm we have found the explicit form of the multiplicities m λ (U k ) in two cases: (i) for the "largest" partitions λ = (λ 1 , . . . , λn) which satisfy λ k+1 + · · · + λn = k − 1; (ii) for the first several k and any λ. (R) is the homogeneous component of degree (n 1 , . . . , n d ) of F d (R), is a symmetric function which plays the role of the character of the corresponding GL d -representation. The Schur functions S λ (

Application of Full Quivers of Representations of Algebras, to Polynomial Identities

Communications in Algebra, 2011

In [7] we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras, and better suited for algebras over finite fields. Here, we consider full quivers as a combinatorial tool in order to describe PI-varieties of algebras. We apply the theory to clarify the proofs of diverse topics in the literature: Determining which relatively free algebras are weakly Noetherian, determining when relatively free algebras are finitely presented, presenting a quick proof for the rationality of the Hilbert series of a relatively free PI-algebra, and explaining counterexamples to Specht's conjecture for varieties of Lie algebras.

Codimensions of algebras and growth functions

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.

On semi-invariants of tilted algebras of type A n

Colloquium Mathematicum

We prove that for algebras obtained by tilts from the path algebras of equioriented Dynkin diagrams of type A n , the rings of semi-invariants are polynomial. Introduction. Let Q = (Q 0 , Q 1) be a quiver with the set Q 0 of vertices and Q 1 of arrows. For every arrow α ∈ Q 1 , we denote by t(α) and h(α) the tail and head of α. Fix an algebraically closed field K of characteristic zero. Let d = (d x) x∈Q 0 be a dimension vector for Q and let V x = K d x for every x ∈ Q 0. The representation variety of the quiver Q in dimension d is the affine variety R(Q, d) = α∈Q 0 Hom(V t(α) , V h(α)). The algebraic group G(d) = x∈Q 0 GL(d x) acts on the variety R(Q, d) in a natural way and the classification problem for representations of Q in dimension d is equivalent to the classification of orbits of that action. The first approximation to the problem is to describe the invariants of G(d) on regular functions on R(Q, d), since the ring of invariants describes closed orbits. But the ring of invariants is trivial unless the quiver Q has oriented cycles. It turns out that one obtains more subtle information by taking regular functions which are invariant with respect to the subgroup G ′ (d) = x∈Q 0 SL(d x) of G(d). The invariants of G ′ (d) are called semi-invariants and we denote the ring of semi-invariants by S(Q, d). In particular it was proven in [12] that one can read off the representation type of a quiver from the algebraic structure of the rings of its semi-invariants. Namely, a quiver Q is of tame representation type if and only if the ring S(Q, d) is a complete intersection for every dimension vector d. For quivers with relations one can repeat the same construction but the situation gets more complicated, since the varieties of representations are no more affine spaces. In that case the research seems to be on the stage of collecting examples. In [5], [11] some rings of semi-invariants were calculated for representation varieties of canonical algebras. The purpose of this paper is to describe the rings of semi-invariants for tilted algebras of type A n. Let Q be a quiver of type A n , let V 1 ,. .. , V n be the indecomposable pairwise nonisomorphic representations of Q, and let

An Upper Bound for the Lower Central Series Quotients of a Free Associative Algebra

International Mathematics Research Notices, 2010

Feigin and Shoikhet conjectured in [FS] that successive quotients Bm(An) of the lower central series filtration of a free associative algebra An have polynomial growth. In this paper we give a proof of this conjecture, using the structure of a representation of Wn, the Lie algebra of polynomial vector fields on C n , on Bm(An) which was defined in [FS]. Moreover, we show that the number of squares in a Young diagram D corresponding to an irreducible Wn-module in the Jordan-Hölder series of Bm(An) is bounded above by the integer (m − 1) 2 + 2[ n−2 2 ](m − 1), which allows us to confirm the structure of B 3 (A 3 ) conjectured in [FS].