arXiv subjects
Publications and source records attributed to Allan Berele.
We show that the path algebra of a quiver satisfies the same polynomial identities of an algebra of matrices, if any. In particular, the algebra of nxn matrices is PI-equivalent to the path algebra of the oriented cycle with n vertices.
A variable line through the centroid G of a triangle divides the triangle into two parts each of whose lengths as a fraction of the perimeter fills a closed interval [m,1-m], with m between 0 and 1/2. We show that the range of m taken over all triangles is the interval (3/10,4/9], with 3/10 approached by scales of the triangles approaching the 5-4-1 triangle and their mid-size medians, and 4/9 attained by the equilateral triangles and the lines through G parallel to the sides. This result is the perimeter version of the classical Winternitz theorem for a triangle, asserting that, in the case of area-ratio instead of perimeter-ratio, m=4/9, and this is attained by all triangles and their lines through G and parallel to the sides.
In [A. Berele, Computing super matrix invariants, {\it Advances in Applied Math. \bf48} (2012), 273--289.] we defined integrals that approximated the Poincaré series of the invariants and concomitants of the general linear Lie supergroup or superalgebra. Budzik suggested in [K. Budzik, Supergroup Invariants and the Brane/Negative Brane Expansion, (preprint) arXiv:2509.20451] a way to adapt this method to get the exact Poincaré series. The purpose of this paper is to prove that Budzik's ideas are correct. As a consequence we prove that the Poincaré series are rational functions.
Given $d_1,\ldots,d_k$ in the field $F$, there is a weighted trace function $F^k\rightarrow F$ given by $tr(x_1,\ldots,x_k)=\sum d_ix_i$. We prove that $F^k$ satisfies trace identities of the forms $α(d_1,\ldots,d_k) x^N=$ a linear combination of terms with lower powers of $x$; and $tr(y_1)\cdots tr(y_n)=$ a linear combination of terms with fewer traces. The approach uses specialized symmetric functions.
We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{σ(1)} \cdots x_{σ(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $σ$ not fixing 1 or $n$ are eventually commutative in the sense that the equality $x_1\cdots x_k = x_{τ(1)} \cdots x_{τ(k)}$ holds for $k$ large enough and all permutations $τ\in S_k$. Calling the minimal such $k$ the degree of eventual commutativity, we prove that $k$ is never more than $2n-3$, and that this bound is sharp. For various natural examples, we prove that $k$ can be taken to be $n+1$ or $n+2$. In the case when $q \ne 1$, we establish that the algebra must be nilpotent. We, moreover, demonstrate that if an algebra is eventually commutative of arbitrary characteristic, then it has a finite basis of its polynomial identities, thus confirming the Specht conjecture in this particular case.
We study the Poincaré series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of $2\times2$ matrices; a denominator, which we presume but have not been able to prove to be in lowest terms, in the case of $3\times3$ matrices; and a conjectured denominator in the general case.
The paper of Domokos is "Invariants of quivers and wreath products," and as in that paper we use Regev's double centralizer theorem for wreath products to study trace identities and invariant theory. In addition to extending to other algebras and groups, we also compute Poincare series and prove embedding theorems.
We generalize the notion of trace identity to $J$-trace. Our main result is that all $J$-traces of $M_{n,n}$ are consequence of those of degree $\frac12n(n + 3)$. This also gives an indirect description of the queer trace identities of $M_n(E)$.
We use hook Schur functions to compute trace cocharacters of matrices in hooks. As an example we compute the cocharacters of 3 x 3 matrices in the one by two hook.
In [Trace identities and $\bf {Z}/2\bf {Z}$-graded invariants, {\it Trans. Amer. Math. Soc. \bf309} (1988), 581--589] we generalized the first and second fundamental theorems of invariant theory from the general linear group to the general linear Lie superalgebra. In the current paper we generalize the computations of the the numerical invariants (multiplicities and Poincaré series) to the superalgebra case. The results involve either inner products of symmetric functions in two sets of variables, or complex integrals. we generalized the first and second fundamental theorems of invariant theory from the general linear group to the general linear Lie superalgebra. In the current paper we generalize the computations of the the numerical invariants (multiplicities and Poincaré series) to the superalgebra case. The results involve either inner products of symmetric functions in two sets of variables, or complex integrals.
In this paper we introduce doubly symmetric functions, arising from the equivalence of particular linear combinations of Schur functions and hook Schur functions. We study algebraic and combinatorial aspects of doubly symmetric functions, in particular as they form a subalgebra of the algebra of symmetric functions. This subalgebra is generated by the odd power sum symmetric functions. One consequence is that a Schur function itself is doubly symmetric if and only if it is the Schur function of a staircase shape.