Proper Central Colengths in PI-Algebras

• Proper central colengths are numerical invariants that count the independent multilinear proper central polynomials modulo identities in associative algebras.
• They link combinatorial properties of Sₙ-modules and character theory with algebraic structures such as gradings and superalgebra constructions.
• Recent classification results leverage these colengths to distinguish varieties of PI-algebras by correlating growth rates with explicit finite lists of minimal generating algebras.

A proper central colength is a numerical and representation-theoretic invariant in the theory of polynomial identities (PI) of associative algebras over a field of characteristic zero. It measures, degree-by-degree, the “abundance” of multilinear proper central polynomials modulo the ordinary identities, and provides a fine-grained tool for classifying varieties of PI-algebras according to the exponential or polynomial growth of these objects. Proper central colengths connect the combinatorial structure of $S_n$-modules arising from the permutation action on the variables, the character theory of symmetric groups, and the structural properties of associative algebras, including gradings and superalgebra constructions. Central to recent advances are the classification theorems linking boundedness or rates of growth of proper central colengths to the inclusion of explicit finite lists of minimal algebras or graded algebras among the generators of a variety.

1. Formal Definitions and Module-Theoretic Framework

Let $F$ be a field of characteristic zero, and $A$ an associative $F$-algebra. The free associative algebra $F\langle X\rangle$ in countably many noncommuting variables $X = \{x_1, x_2, \dots\}$ serves as the ambient algebra for polynomial identities. The T-ideal of identities is $$\mathrm{Id}(A) = \{ f \in F\langle X\rangle : f(a_1,\ldots,a_m) = 0 \text{ for all } a_i\in A \}$$. The T-space of central polynomials is $$\mathrm{Id}^z(A) = \{ f \in F\langle X\rangle : f(a_1,\ldots, a_m) \in Z(A) \text{ for all } a_i\in A\}$$.

For multilinear analysis, set $$P_n = \mathrm{span}_F\{x_{\sigma(1)}\cdots x_{\sigma(n)} : \sigma\in S_n\}$$ and define the relevant $S_n$-modules:

• $P_n(A) = P_n / (P_n \cap \mathrm{Id}(A))$,
• $P_n^z(A) = P_n / (P_n \cap \mathrm{Id}^z(A))$,
• $$\Delta_n(A) = (P_n \cap \mathrm{Id}^z(A)) / (P_n \cap \mathrm{Id}(A))$$.

The module $\Delta_n(A)$ is the space of multilinear proper central polynomials modulo identities. Each $S_n$-module decomposes as a direct sum of irreducible Specht modules $S^\lambda$ indexed by partitions $\lambda \vdash n$. The sequence of proper central colengths $(\ell_n^\delta(A))$ is given by

$$\ell_n^\delta(A) = \dim_F \Delta_n(A) = \sum_{\lambda \vdash n} m_{n,\lambda}^\delta(A),$$

where $m_{n,\lambda}^\delta(A)$ is the multiplicity of $S^\lambda$ in the decomposition of $\Delta_n(A)$.

2. Proper Central Polynomials, Colengths, and Exponents

A central polynomial $f \in \mathrm{Id}^z(A)$ is proper if $f \notin \mathrm{Id}(A)$. The proper central colength $\ell_n^\delta(A)$ (also called the proper central codimension in some literature) counts, up to the $S_n$-action, the number of independent multilinear proper central polynomials in $n$ variables.

For a variety of associative algebras $\mathcal V$, the corresponding sequence $c_n^\delta(\mathcal V)$ is defined analogously as the dimension of the quotient of multilinear central polynomials modulo identities, taken in the relatively free algebra of the variety.

The asymptotic growth is quantified by the proper central exponent: $$\exp_\delta(\mathcal V) = \limsup_{n\to\infty} (c_n^\delta(\mathcal V))^{1/n}.$$ Giambruno–Zaicev and subsequent works established that $\exp_\delta(\mathcal V)$ exists as an integer and that either $c_n^\delta(\mathcal V)$ grows polynomially (then $\exp_\delta \le 1$) or exponentially with integer base.

3. S_n-Module Decomposition, Multiplicities, and the Highest-Weight Test

For each degree $n$, the modules $P_n(A)$, $P_n^z(A)$, and $\Delta_n(A)$ decompose into Specht modules: $$P_n(A) \cong \bigoplus_{\lambda \vdash n} m_{n,\lambda}(A) S^\lambda, \quad P_n^z(A) \cong \bigoplus_{\lambda \vdash n} m_{n,\lambda}^z(A) S^\lambda, \quad \Delta_n(A) \cong \bigoplus_{\lambda \vdash n} m_{n,\lambda}^\delta(A) S^\lambda,$$ with $$m_{n,\lambda}(A) = m_{n,\lambda}^z(A) + m_{n,\lambda}^\delta(A)$$. The characters satisfy

$\chi_n(A) = \chi_n^z(A) + \chi_n^\delta(A).$

To determine $m_{n,\lambda}^\delta(A) > 0$, one applies the highest-weight-vector test: given a tableau $T_\lambda$ for $\lambda\vdash n$, construct

$$f_{T_\lambda} = \left(\prod_{i=1}^{\lambda_1} \mathrm{St}_{h_i(\lambda)}(x_1,\ldots,x_{h_i(\lambda)})\right) \sigma_{T_\lambda}^{-1}$$

with $\mathrm{St}_t$ the standard polynomial of degree $t$ and $\sigma_{T_\lambda}$ the permutation to $T_\lambda$. Then $m_{n,\lambda}^\delta(A) > 0$ if and only if $f_{T_\lambda} \notin \mathrm{Id}(A)$ but $f_{T_\lambda} \in \mathrm{Id}^z(A)$. The determination of dimensions and multiplicities relies on hook-length formulas and character-theoretic arguments (Cota et al., 14 Jan 2026).

4. Explicit Examples: Matrix, Grassmann, and Block Algebras

The proper central colength sequence varies dramatically with the structure of $A$:

Algebra	$\ell_n^\delta(A)$	Main Features
$M_d(F)$, $UT(d)$	$0$	No proper central polynomials; $T^z(A) = T(A)$.
Infinite Grassmann $\mathcal G$	$n-1$	$c_n(\mathcal G) = 2^{n-1}$, $c_n^z(\mathcal G) = 2^{n-2}$, $$\chi_n^\delta(\mathcal G) = \sum_{i=0}^{n-2}\chi_{(n-2-i,1^i)}$$ (Regev, 2015).
Finite Grassmann $\mathcal G_{2k}$	$k+1$	$\ell_n(\mathcal G_{2k}) = 2k+1$, $\ell_n^z(\mathcal G_{2k}) = k$.
$UT_2$	$0$	No proper central polynomials.

For full matrix algebras $M_k(F)$, the asymptotic proper central colength grows as $k^2$: $$\lim_{n \to \infty} (\text{on}_n(M_k(F)))^{1/n} = k^2 \qquad [1504.07077].$$ For the Grassmann algebra $\mathcal G$,

$$\ell_n^\delta(\mathcal G) = n-1, \qquad c_n^\delta(\mathcal G) = 2^{n-2}.$$

This exponential richness of proper central polynomials (contrasting the triviality in $M_d(F)$) is fundamental to central PI-theory (Regev, 2015, Cota et al., 14 Jan 2026).

5. Classification by Asymptotic Growth: Minimal Algebras and Exponent Theory

Recent work achieves full classification of varieties based on the growth rate of proper central colengths. Benanti–Valenti (Benanti et al., 2024, Benanti et al., 12 May 2025) constructed minimal lists of finite-dimensional algebras such that

• $\exp_\delta(\mathcal V) > 2$ if and only if some $A_i$ lies in $\mathcal V$ (for the $9$ explicit algebras in the ungraded case, and $12$ in the $G$-graded case).
• $\exp_\delta(\mathcal V) = 2$ precisely when these $A_i$ are excluded, but at least one of the “almost polynomial” growth generators ($\mathcal G, D, D_0$) is present.

No intermediate growth rates appear: proper central exponents jump in integer steps (e.g., $2,3,4,p$), fully determined by the minimal centrally admissible semisimple subalgebras present in a Grassmann envelope (Kemer theory).

The key proof ingredients are:

• Reduction to finite-dimensional superalgebras $B = B_{\rm ss} \oplus J(B)$;
• Identification of centrally admissible subalgebras dictating the exponent;
• Analysis of block-triangular and Grassmann-envelope algebras;
• Explicit computations of proper central colengths via combinatorial representation theory (Benanti et al., 2024, Benanti et al., 12 May 2025).

6. Combinatorial and Representation-Theoretic Methods

The computation of proper central colengths leverages advanced combinatorial and representation-theoretic tools:

• The Drensky–Berele “multilinear–GL-module” approach, using Schur–Weyl duality to relate multiplicities of Specht modules in $GL_m$-modules to $S_n$-modules;
• Highest-weight vector techniques, with construction of test polynomials from tableaux shapes and standard polynomials;
• Hook-length formulas, controlling both the dimension of Specht modules and the asymptotics of codimension sequences;
• Character-theoretic formulas for inner products $\langle\chi_n^\delta, \chi_\lambda\rangle$ to extract multiplicities.

In the graded setting, these methods generalize through consideration of multidegree and group grading, leading to analogous exponents and classification theorems (Benanti et al., 12 May 2025).

7. Open Questions and Further Directions

Open problems include determining the precise polynomial factors in the asymptotic expressions $c_n^\delta(A) \sim a\, n^\alpha\, d^n$ for simple or semisimple algebras (e.g., finding $\alpha$ for $M_k(F)$), and the finite Specht property for the algebra of central polynomials over the identity ideal for general PI-algebras (Regev, 2015). Another domain of investigation is the extension of exponent theory and classification to group-graded PI-algebras and analysis of gradings, superalgebras, and non-associative variants. The combinatorial and character-theoretic approach provides the blueprint for these generalizations.

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References:

• (Cota et al., 14 Jan 2026)
• (Regev, 2015)
• (Benanti et al., 2024)
• (Benanti et al., 12 May 2025)