A structural classification of algebras with graded involution and quadratic codimension growth

Technical Summary: A Structural Classification of Algebras with Graded Involution and Quadratic Codimension Growth

Problem Statement
The paper addresses a central problem in the theory of algebras with polynomial identities (PI-theory): the classification of varieties according to the asymptotic behavior of their codimension sequences. While the linear growth case has been fully understood and the exponential growth case characterized by the presence of specific algebras (the Grassmann algebra $G$ and the algebra of $2 \times 2$ upper triangular matrices $UT_2$), the general classification of varieties with quadratic growth remains a significant open challenge.

Specifically, the authors focus on unitary associative $G$-graded algebras endowed with a graded involution (termed $(G, \ast)$-algebras), where $G$ is a finite group. The goal is to provide a complete classification, up to $T^*_G$-equivalence, of unitary varieties with quadratic codimension growth. A key open question motivating this work is whether every variety of polynomial growth is generated, up to equivalence, by a direct sum of algebras generating minimal varieties.

Methodology
The authors employ a structural approach rooted in the representation theory of symmetric groups and the theory of proper polynomials. The methodology proceeds as follows:

1. Proper Cocharacters: The study utilizes the space of proper $(G, \ast)$-polynomials, $\Gamma^\sharp_n$, which generate the ideal of identities for unitary algebras. The codimension sequence $c^\sharp_n(A)$ is related to the proper codimensions $\gamma^\sharp_n(A)$ via a binomial sum.
2. Multiplicity Analysis: The authors decompose the proper cocharacters $\pi_{n_1, \dots, n_{2k}}(A)$ into irreducible characters of the product of symmetric groups $S_{n_1} \times \dots \times S_{n_2k}$. The core of the analysis involves determining the nonzero multiplicities ($m_{\lambda_1, \dots, \lambda_{2k}}$) in this decomposition.
3. Minimal Varieties as Building Blocks: The paper relies on a prior classification of minimal varieties of quadratic growth (by Cota and Vieira [7] and others). These minimal varieties are generated by specific finite-dimensional algebras, including:
   • Commutative subalgebras of upper triangular matrices ($C_m$) with specific gradings and involutions.
   • Subalgebras of upper triangular matrices ($N_m, U_m$) with elementary gradings.
   • Subalgebras of the Grassmann algebra ($G_2$) with various involutions ($\psi, \tau, \gamma$).
   • Specific commutative subalgebras of $UT_4$ ($W$) with defined involutions.
4. Correspondence Theorem: The authors establish a direct correspondence between the nonzero multiplicities in the decomposition of the proper cocharacters of an algebra $A$ and the specific minimal varieties contained within the variety generated by $A$.
5. Direct Sum Decomposition: By analyzing the multiplicities for degrees $n=1$ and $n=2$ (since quadratic growth implies vanishing proper identities for $n \ge 3$), the authors construct a direct sum of minimal algebras that is $T^*_G$-equivalent to the original algebra.

Key Contributions and Results

• Classification of Minimal Varieties: The paper consolidates and utilizes the classification of minimal unitary $(G, \ast)$-varieties of quadratic growth. It explicitly lists the $T^*_G$-ideals, codimension sequences, and proper cocharacters for a comprehensive set of generating algebras (denoted as set $\mathcal{I}$ in the text).
• Multiplicity-Structure Correspondence: The primary theoretical contribution is the proof that for a unitary $(G, \ast)$-algebra $A$ with quadratic growth, the nonzero multiplicities in its proper cocharacter decomposition uniquely determine the minimal varieties contained in $\text{var}^\sharp(A)$. Specifically:
  • Multiplicities of type $m_{((1)h^+)}$ correspond to the presence of $C^h_2$.
  • Multiplicities of type $m_{((2)g^-)}$ correspond to $C^g_{3,\ast}$.
  • Multiplicities of type $m_{((1,1)1^+)}$ correspond to $U^1_{3,\ast}$.
  • Multiplicities of type $m_{((1)s^\epsilon, (1)t^\delta)}$ (equal to 1 or 2) correspond to specific sums of algebras like $G_{s,t} \oplus W_{s,t}$ or individual algebras like $W_{s,t}$, depending on the value of the multiplicity.
• Main Theorem (Theorem 4.10): The paper proves that any finite-dimensional unitary $(G, \ast)$-algebra $A$ with quadratic codimension growth is $T^*_G$-equivalent to a finite direct sum of algebras from the set $\mathcal{I}$. Crucially, at least one component in this sum must belong to the subset $\mathcal{I} \setminus \{F, C^h_2, C^g_{2,\ast}\}$ (the algebras generating strictly quadratic growth).
• Corollary 4.11: The authors conclude that a finite-dimensional unitary $(G, \ast)$-algebra has codimension growth bounded by $O(n^2)$ if and only if it is $T^*_G$-equivalent to a finite direct sum of algebras generating minimal varieties with at most quadratic growth.

Significance
The paper claims that its results constitute a significant step toward a comprehensive classification of varieties of quadratic codimension growth in the context of graded algebras with involution. By establishing that every such variety is generated by a direct sum of minimal varieties, the work provides a structural decomposition that mirrors the known results for linear growth and ordinary PI-algebras. The approach bridges the gap between the abstract theory of codimension growth and the concrete structural properties of the generating algebras, offering a complete characterization for the unitary case. The results are presented as intrinsic to PI-theory, extending the classification framework to algebras with additional structures (grading and involution).