Related papers: Multivariate Poincar\'e series for algebras of $SL…
Let $\mathcal{I}_{d_1,d_2}$ and $\mathcal{C}_{d_1,d_2}$ be the algebras of joint invariants and joint covariants of the two binary forms of degrees $d_1$ and $d_2.$ Formulas for computation of the Poincar\'e series…
A formula for computation of the bivariate Poincar\'e series $\mathcal{P}_d(z,t)$ for the algebra of covariants of binary $d$-form is found.
Explicit formulas for computation of the Poincar\'e series for the algebras of joint $SL_2$-invariants and covariants of $n$ linear forms in terms of Narayana polynomials are found. Also, for these algebras we calculate the degrees and…
A formula for computation of the Poincar\'e series $P_d(z)$ of the algebra of the covariants of binary $d$-form is found. By using it, we have computed the $P_d(z)$ for $d \leq 20.$
We offer a Maple package {\tt Poincare\_Series} for calculating the Poincar\'e series for the algebras of invariants/covariants of binary forms, for the algebras of joint invariants/covariants of several binary forms, for the kernel of…
The formula for the Poincare series of the algebra of invariant of $n$-ary form is found.
Analogue of Springer's formula for the Poincar\'e series of the algebra invariants of ternary form is found.
We propose a heuristic algorithm for fast computation of the Poincar\'{e} series $P_n(t)$ of the invariants of binary forms of degree $n$, viewed as rational functions. The algorithm is based on certain polynomial identities which remain to…
Let (N, G), where N is a normal subgroup of G<SL_n(C), be a pair of finite groups and V a finite-dimensional fundamental G-module. We study the G-invariants in the symmetric algebra S(V) by giving explicit formulas of the Poincar\'{e}…
A hypergraph $H=(V,E)$, where $V=\{x_1,...,x_n\}$ and $E\subseteq 2^V$ defines a hypergraph algebra $R_H=k[x_1,...,x_n]/(x_{i_1}... x_{i_k}; \{i_1,...,i_k\}\in E)$. All our hypergraphs are $d$-uniform, i.e., $|e_i|=d$ for all $e_i\in E$. We…
Many invariants of finitely generated positive cancelative commutative semigroups can be studied from their Poincar\'e series. We offer and present several closed formulas for them. Moreover, those formulas have elementary proofs and are…
By using MacMahon partition analysis technique, the Poincar\'e series for the algebras of invariants of the ternary, quaternary and quinary forms of small orders are calculated.
Earlier, there was computed the Poincar\'e series of a valuation or of a collection of valuations on the ring of germs of holomorphic functions in two variables. For a collection of several plane curve valuations it appeared to coincide…
In this note, we study the Hilbert-Poincar\'e polynomials for the PBW-graded of simple modules for a simple complex Lie algebra. The computation of their degree can be reduced to modules of fundamental highest weight. We provide these…
We calculate the degree of the algebra of covariants $\mathcal{C}_d$ for binary $d$-form. Also, for the degree we obtain its integral representation and asymptotic behavior.
In this paper, we discuss the Poincar\'{e} series of Kac-Moody Lie algebras, especially for indefinite type. Firstly, we compute the Poincar\'{e} series of certain indefinite Kac-Moody Lie algebras whose Cartan matrices have the same type…
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…
For a subfield $\K$ of the field $\C$ of complex numbers, we consider curve and divisorial valuations on the algebra $\K[[x,y]]$ of formal power series in two variables with the coeficients in $\K$. We compute the semigroup Poincar\'e…
Let $N$ be a normal subgroup of a finite group $G$ and $V$ be a fixed finite-dimensional $G$-module. The Poincar\'{e} series for the multiplicities of induced modules and restriction modules in the tensor algebra $T(V)=\oplus_{k \geq…
Functional bases of second-order differential invariants of the Euclid, Poincar\'e, Galilei, conformal, and projective algebras are constructed. The results obtained allow us to describe new classes of nonlinear many-dimensional invariant…