Shift invariant algebras, Segre products and regular languages (Q6169051)

The equivariant Hilbert series of a family of noetherian standard algebras \({A} = (A_n)_{n\in\mathbb{N}}\) is defined as a formal power series in two variables \(s\) and \(t\): \[ \equiv H_A(t, s) :=\Sigma_{n\geq 1}^{j\geq 0}\dim_K[A_n]_jt^js^n. \] The problem considered here is establishing instances where this series is a rational function. The tool to study these algebras and their Hilbert function is to relate them to certain regular formal languages; a formal language is given by a subset of the words in a given finite alphabet; and a language is said to be regular if it can be realized by a finite automaton (a directed graph whose vertices are labelled by exactly all the word of the language and whose edges represent transitions). A weight function on a language \(\mathcal{L}\) is a monoid homomorphism \(\rho: \mathcal{L} \rightarrow \mathrm{Mon}(T)\), where \(\mathrm{Mon}(T)\) denotes the set of monomials of a polynomial ring \(T\) in finitely many variables \(s_1,\dots, s_k\). Its generating function is a formal power series \[ P_{L,\rho}(s1,\dots, sk) = \sum_{w\in L}\rho(w). \] It is known that if \(\mathcal{L}\) is a regular language, this series is a rational function (that can be computed explicitely via an automaton realizing the language). In the paper, languages \(\mathcal{L}_\mathcal{A}\) on an alphabet \(\Sigma_\mathcal{A} = {\tau_{1,1},\dots, \tau_{1,a}, \alpha_1,\dots,\alpha_p}\) are considered. Every word in \(\mathcal{L}_\mathcal{A}\) can be written as \(\tau_1^{k_1}\alpha_{i_1}\tau_1^{k_2}\alpha_{i_2}\ldots \tau_1^{k_{d+1}}\), with integers \(d \geq 0\), \(1 \leq i_1, \ldots i_d \leq p\), and \(\tau_1^{k_l}\) is some string that uses only letters \(\tau_{1,1}, \dots, \tau_{1,a}\). Words in \(\mathcal{L}_\mathcal{A}\) are words of the form above with conditions on \(k_1,\dots, k_{d+1}\) and \(i_1,\dots, i_d\). If we have two such languages \(\mathcal{L}_\mathcal{A}\), \(\mathcal{L}_\mathcal{B}\), their Segre product \(\mathcal{L}_\mathcal{A}\boxtimes\mathcal{L}_\mathcal{B}\) is defined by words of the form \(\tau_1^{k_1}\tau_2^{l_1}\gamma_{i_1,j_1}\dots \tau_1^{k_d}\tau_2^{l_d}\gamma_{i_d,j_d} \tau_1^{k_{d+1}}\tau_2^{l_{d+1}}\) where \(\tau_1^{k_1}\alpha_{i_1}\tau_1^{k_2}\alpha_{i_2}\ldots \tau_1^{k_{d+1}}\in\mathcal{L}_\mathcal{A}\) and \(\tau_2^{l_1}\beta_{j_1}\tau_2^{l_2}\beta_{j_2}\ldots \tau_2^{l_{d+1}}\in\mathcal{L}_\mathcal{B}\). A key result in the paper is that if \(\mathcal{L}_\mathcal{A}\),\(\mathcal{L}_\mathcal{B}\) are regular, then their Segre product is again a regular language. As an application, the rationality of equivariant Hilbert series in new cases is established (regular languages have been used previously to prove such rationality results). For example, Segre products of filtrations whose factors are represented by regular languages have a rational equivariant Hilbert series. Since filtrations determined by Segre products of filtrations of algebras are considered, the above theorem gives (Theorem 3.7) that every filtration of algebras given as a tensor product of families of algebras with rational equivariant Hilbert series has a rational equivariant Hilbert series.