Asymptotics for the standard and the Capelli identities

Asymptotics for Amitsur's Capelli-type polynomials and verbally prime PI-algebras

“…By applying the Littlewood -Richardson rule, Berele and Regev in [2, Theorem 1.1] give a rule to calculate the n-th cocharacter of a products of T -ideals. By this rule and by the results about the form of the n-th cocharacter of the verbally prime algebras mentioned before, similar to [3,9] we can estimate the size of a hook and a square containing all diagrams which appear with non-zero multiplicity in the decomposition of the n-th cocharacter of T (G(C)). More precisely…”

“…Also in [9] it was proved that the codimensions of Γ k 2 ,0 are asymptotically equal to the codimensions of the verbally prime algebra M k (F )…”

Algebras with involution with linear codimension growth

“…Hence Id * (U 1 ⊕ U 2 ) ⊆ Id * (B). On the other hand, the subalgebras F + J 11 and F + J 01 + J 10 are isomorphic to the algebras U 1 and U 2 , respectively. Hence Id * (B) ⊆ Id * (U 1 ) ∩ Id * (U 2 ) = Id * (U 1 ⊕ U 2 ) and equality holds.…”

“…Recall the following result given in [11,Lemma 2] on the decomposition of the Jacobson radical of a finite-dimensional algebra. Notice that if the algebra A has an involution, then in the above lemma J 00 and J 11 are stable under the involution whereas J * 10 = J 01 .…”

Characterizing Varieties of Colength ≤4

“…Let A = F + J be a finite dimensional algebra over F where J is the Jacobson radical of A, then J can be decomposed into the direct sum of F -bimodules (see for instance Giambruno and Zaicev, 2003), i.e., J = J 00 + J 01 + J 10 + J 11 , where for i ∈ 0 1 J ik is a left faithful module or a 0-left module according as i = 1 or i = 0 respectively. Similarly, J ik is a right faithful module or a 0-right module according as k = 1 or k = 0 respectively.…”