Group graded algebras and varieties with quadratic codimension growth

Let $A$ be an associative algebra graded by a finite group $G$ over a field ${F}$ of characteristic zero. One associates to $A$ the sequence of $G$-graded codimensions $c_n^G(A)$, $n=1,2,\ldots$, which measures the growth of the polynomial identities satisfied by $A$. It is known that this sequence is either polynomially bounded or grows exponentially. In this paper, we study unitary $G$-graded varieties of polynomial codimension growth. In particular, we classify the varieties generated by unitary algebras with quadratic codimension growth and show that these varieties can be described as a direct sums of algebras that generate minimal $G$-graded varieties.

💡 Research Summary

The paper investigates the growth of G‑graded codimensions for associative algebras A over a field of characteristic zero, where G is a finite group. The G‑graded codimension sequence c⁽ᴳ⁾ₙ(A) measures the dimension of multilinear G‑graded polynomials modulo the T_G‑ideal of G‑graded identities of A. Building on the classical Regev–Kemer dichotomy, which states that ordinary codimensions are either polynomially bounded or exponential, the authors extend this dichotomy to the graded setting and focus on varieties whose graded codimensions grow polynomially, especially those with quadratic growth (degree two).

The authors first recall the construction of the free G‑graded algebra F⟨X,G⟩, the multilinear spaces P⁽ᴳ⁾ₙ, and the definition of proper G‑graded polynomials—linear combinations of monomials where the neutral element of G appears only inside long Lie commutators. They show that for unitary algebras the T_G‑ideal is generated by multilinear proper graded polynomials, and they relate the ordinary graded codimensions c⁽ᴳ⁾ₙ(A) to the proper graded codimensions γ⁽ᴳ⁾ₙ(A) via the binomial identity c⁽ᴳ⁾ₙ(A)=∑_{i=0}^{n} (n choose i) γ⁽ᴳ⁾_i(A). Consequently, if the codimension sequence is polynomially bounded, γ⁽ᴳ⁾_m(A)=0 for all m larger than the degree of the polynomial.

A key structural result (Theorem 2.1, due to La Mattina) characterizes algebras with polynomially bounded graded codimensions: they are T_G‑equivalent to a finite direct sum of finite‑dimensional G‑graded algebras whose semisimple part has dimension at most one. This reduces the classification problem to studying a finite family of “building blocks”.

The paper then develops the representation‑theoretic machinery needed to analyze proper graded cocharacters. For a fixed composition n=n₁+…+n_k (where k=|G|), the space of multilinear proper polynomials Γ_{n₁,…,n_k} carries an action of the product symmetric group S_{n₁}×…×S_{n_k}. Its character decomposes into outer tensor products of irreducible S_{n_i}‑characters indexed by multipartitions (λ₁,…,λ_k). The multiplicities m_{λ₁,…,λ_k} are shown to equal the maximal number of linearly independent proper highest‑weight vectors of the given multipartition that survive modulo Id⁽ᴳ⁾(A). This connection allows the authors to compute the graded cocharacters for low degrees (n=1,2) explicitly, providing a table of proper cocharacters and the associated highest‑weight vectors.

With this machinery, the authors construct explicit families of minimal G‑graded varieties of quadratic codimension growth. For each group element g of order two and each integer m≥2 they define a commutative G‑graded algebra C_{g,m} inside the upper triangular matrix algebra UT_m, assigning degree 1 to the identity matrix and degree g to the first sub‑diagonal matrices. They also define a 7‑dimensional algebra K_{g,h}⁷ for distinct non‑identity elements g,h∈G, with a specific grading on the matrix units. Lemma 4.1 gives the G‑graded identities of these algebras and computes their graded codimension sequences: c⁽ᴳ⁾ₙ(C_{g,m}) = Σ_{i=0}^{m-1} (n choose i) and c⁽ᴳ⁾ₙ(K_{g,h}⁷) = 1 + 2n + (n choose 2). Both are quadratic polynomials in n, confirming that the varieties they generate have exactly quadratic growth.

The main classification theorem states that any unitary G‑graded variety with quadratic graded codimension growth is a direct sum of varieties generated by algebras of the above two types (or their T_G‑equivalents). In other words, every such variety decomposes as V ≅ ⊕{i∈I} var_G(A_i), where each A_i is either some C{g,m} or some K_{g,h}⁷, and the index set I may be finite or infinite depending on the structure of G. This mirrors the known classification for Z₂‑graded algebras (reference