Bounds on colength and maximal multiplicity sequences.
Let \(A\) be an associative algebra over a field of characteristic 0. In order to study the ideal \(T(A)\) of the polynomial identities satisfied by \(A\) it is sufficient to study the multilinear ones. The vector space \(P_n\) of the multilinear polynomials in \(x_1,\dots,x_n\) in the free associative algebra is a module over the symmetric group \(S_n\) acting by permuting the variables; in fact \(P_n\) is isomorphic to the regular module for \(S_n\). The multilinear identities for \(A\) in \(P_n\) form a submodule. It is well known that if \(A\) satisfies a nontrivial identity then \(T(A)\cap P_n\) becomes huge when \(n\to\infty\), so one is led to study the quotient \(P_n(A)=P_n/(P_n\cap T(A))\). There are several important numerical characteristics of the identities of \(A\) associated to the modules \(P_n(A)\). Probably the most widely studied is the codimension sequence of \(A\), \(c_n(A)=\dim P_n(A)\). The representations of \(S_n\) are described in terms of partitions and Young diagrams. Let \(P_n(A)=\oplus m_\lambda M(\lambda)\) be the decomposition of \(P_n(A)\) into irreducibles \(M(\lambda)\) with multiplicities \(m_\lambda\) where \(\lambda\) runs over the partitions of \(n\). The character \(\chi(A)=\chi(P_n(A))=\sum m_\lambda\chi^\lambda\) is called the \(n\)-th cocharacter of \(A\). Denote by \(\text{mlt}(A)\) the least number \(e\) such that all multiplicities \(m_\lambda\) are bounded above by a polynomial in \(n\) of degree \(e\). The \(n\)-th colength of \(A\) is defined as the sum \(\sum m_\lambda\), and \(\text{col}(A)\) is the least \(e\) such that the \(n\)-th colength of \(A\) is bounded by a polynomial of degree \(e\) in \(n\). Recall also that the exponent of \(A\), \(\exp(A)\), is the least \(e\) such that \(c_n(A)\) is bounded by \(f(n)e^n\) where \(f\) is a polynomial. In order to state the main results of the paper under review one needs the notion of a hook. Let \(a\) and \(b\) be integers, the hook \(H(a,b)\) is the set of all partitions that have at most \(a\) parts of length \(\geq b\). It is well known that if \(A\) satisfies a nontrivial identity then \(\chi(A)\), the cocharacter of \(A\), is supported by an appropriate hook \(H(a,b)\). In other words all \(m_\lambda=0\) whenever \(\lambda\notin H(a,b)\). Denote further \(H(a,b,N)\) the set of all partitions \(\lambda\) such that by removing at most \(N\) boxes from the corresponding diagram one gets a diagram whose partition lies in \(H(a,b)\). The main result of the paper under review is the following. Assume \(A\) is PI and assume the cocharacter of \(A\) is supported by \(H(a,b,N)\) for some \(a\), \(b\), \(N\). Then \(\text{mlt}(A)\leq {a\choose 2}+{b\choose 2}\). Also \(\text{col}(A)\leq {a+1\choose 2}+{b+1\choose 2}-1\). The T-prime algebras play a prominent role in PI theory. The exact values of \(\text{col}(A)\) and of \(\text{mlt}(A)\) are well known for \(A\) a T-prime algebra, see these values in Theorem 3.1 of the paper. The author gives a formula for computing the values of \(\text{col}(A)\) and \(\text{mlt}(A)\) in the case the T-ideal of \(A\) is the product of T-prime T-ideals. Relations among \(\exp(A)\), \(\text{mlt}(A)\) and \(\text{col}(A)\) are given for such algebras as well. Finally the author gives similar relations for the case of a general PI algebra. (Only the lower bound for \(\text{col}(A)\) in the general case is weaker compared to the product of T-primes.)
scientific article; zbMATH DE number 5860490