Dimension Growth in Simple Modules

• The paper presents explicit formulas for the dimension and growth exponent of simple modules, using invariants like the Gelfand–Kirillov dimension.
• It employs combinatorial and geometric methods, including root system analysis and Kazhdan–Lusztig cell filtrations, to classify growth patterns.
• The findings have practical implications in quantum group representations, modular Lie theory, and noncommutative algebraic geometry.

Dimension growth of simple modules refers to the asymptotic behavior, explicit formulas, and possible integer values of the dimension (or polynomial growth exponent) for families of simple modules over associative algebras and related categories. This phenomenon has deep connections with the structure theory of the underlying algebra, the combinatorics of root systems or poset/lattice data, and geometric representation theory via invariants such as the Gelfand–Kirillov dimension or growth exponents. It has been the focus of extensive research across Lie theory, quantum groups, modular representation theory, finite group and algebroid algebras, and noncommutative algebraic geometry.

1. Formalism and Definitions

The growth of a family of simple modules is typically captured by invariants such as:

• Dimension polynomial: Let $A$ be a parametrized algebra and $L_\Lambda$ a simple module depending on a parameter (e.g., a highest weight or deformation scalar). The function $f_\Lambda(\ell) = \dim L_\Lambda$ is often polynomial for large $\ell$.
• Growth exponent: One considers

$$d_{\mathrm{growth}}(\Lambda) = \limsup_{\ell \to \infty} \frac{\log \dim L_\Lambda(\ell)}{\log \ell},$$

which measures polynomial degree.

• Gelfand–Kirillov (GK) dimension: For a finitely generated module $M$ over an algebra $A$, with generating subspaces $V \subset A$, $M_0 \subset M$,

$$\operatorname{GKdim}(M) = \limsup_{n\to\infty} \frac{\log \dim(M_0 V^n)}{\log n}.$$

This quantifies the asymptotic growth of the module (Gupta, 2011, Futorny et al., 16 Dec 2025).

These invariants serve as proxies for complexity and size, generalizing classical dimension where simple modules are finite-dimensional.

2. Dimension Growth in Quantum Groups and Lie Theory

The interplay of dimension growth and GK dimension is particularly rich in quantum group representation theory. For the De Concini–Kac quantum group $U_\zeta$ at a root of unity $\zeta = e^{2\pi i/\ell}$, one considers the family of simple modules $L_\zeta(\Lambda)$ indexed by weights. The growth exponent

$$d_{\mathrm{growth}}(\Lambda) = \limsup_{\ell \to \infty} \frac{\log \dim_{\mathbb{C}} L_\zeta(\Lambda)}{\log \ell}$$

coincides with the GK dimension of the corresponding highest weight module $L_q(\Lambda)$ over the generic quantum group $U_q$ (with $q$ not specialized), as proved by Shapovalov-form and asymptotic matching arguments. More formally,

$$d_{\mathrm{growth}}(\Lambda) = \operatorname{GKdim}_{U_{q}} L_q(\Lambda).$$

This result allows the transfer of combinatorial and geometric information between the two settings (Futorny et al., 16 Dec 2025).

For classical Lie algebras in positive characteristic, the dimensions of irreducible $U_{\lambda,\chi}$-modules (where $\lambda$ is a regular rational central character and $\chi$ a $p$-character) are given by explicit polynomials in $p$, the degree of which corresponds to the cell filtration index of an associated canonical basis element in $K_0$ of the Springer fiber. Thus, the growth rate is governed by the two-sided Kazhdan–Lusztig cell containing the representation (Bezrukavnikov et al., 2017).

3. Characteristic Patterns and Minimal Growth

For semisimple Lie algebras and quantum groups, the possible values of minimal nonzero dimension growth and GK dimension stratify cleanly by Lie type. For each type, the minimal nonzero value is computed via Lusztig’s $a$-function and root system combinatorics. Key results include:

Type	Minimal Quantum $d_{\mathrm{growth}}$	Minimal Classical $d_{\mathrm{growth}}$
$A_n$	$n$	$n$
$B_n$	$n$	$2n-2$
$C_n$	$n$	$n$
$D_n$	$2n-3$	$2n-2$
$E_6$	$11$	$11$
$E_7$	$17$	$17$
$E_8$	$29$	$29$
$F_4$	$8$	$8$
$G_2$	$3$	$3$

A key phenomenon is the strict drop in minimal dimension growth in type $B_n$ for quantum groups compared to the classical case, driven by the new contributions of non-integral weight blocks (Futorny et al., 16 Dec 2025).

In modular Lie theory, the degree of the dimension polynomial of irreducible modules is controlled by the position of the associated canonical basis element in a Kazhdan–Lusztig cell filtration, with degrees equal to cell index and leading coefficients encoding geometric multiplicities (Bezrukavnikov et al., 2017).

4. Families of Algebras: Quantum Tori and Differential Operators

For noncommutative algebras such as quantum Laurent polynomial algebras and skew polynomial rings, dimension growth exhibits rigidity and characteristic gaps:

• For $n$-dimensional quantum tori, possible GK dimensions of simple modules form a "characteristic set" of integers between 1 and $n$, controlled by the parameter matrix and central subalgebra ranks. Dichotomy theorems show that, for Krull dimension $n-1$, only $1$ and $n-2$ may arise (Gupta et al., 2018).
• In the case of simple differential–operator rings $K_n[x;\delta]$ with $\delta$ suitably generic, every simple module has GK dimension either $1$ or $n$—no intermediate values appear (Gupta et al., 2016). This reflects maximal constraints imposed by noncommutativity and Ore-structure.

Tensor products of modules over multiparameter quantum tori are classified via the Brookes–Groves invariant, and GK dimension is additive under external tensor product:

$$\operatorname{GKdim}(M_1 \otimes_F M_2) = \operatorname{GKdim}(M_1) + \operatorname{GKdim}(M_2)$$

(Gupta, 2011).

5. Asymptotic Growth in Families of Combinatorial and Finite Algebras

Dimension growth of simple modules in combinatorial settings can be exponential, polynomial, or bounded, depending on the structure:

• For Boolean matrix algebras (monoids of all binary relations on an $n$-element set), dimensions of simple modules are given by explicit inclusion–exclusion–type formulas:

$$\dim_k S_{E,R,V}(X) = \frac{\dim_k V}{|\mathrm{Aut}(E,R)|} \sum_{i=0}^{|E|} (-1)^i \binom{|E|}{i} (|G| - i)^{|X|}$$

where $G$ is a lattice-theoretic parameter. For antichains, this yields $|G| = 2^{|E|}$, and thus, simple module dimensions can grow as fast as $2^{|E|n}$ for fixed $|E|$, i.e., exponentially in $n$ (Bouc et al., 2019).

• For Temperley–Lieb algebras and related objects, dimensions follow deformed Pascal-type recursions (tilting theory), but for each fixed module label the exponential rate $2^n$ is universal, with only a polynomial (e.g., $n^{-3/2}$) correction. There are no "wild" growth rates depending on deforming parameters or characteristic; all leading exponential terms are robust (Andersen, 2017).

For finite groups in characteristic $p$, lower bounds for the maximal dimension of a simple module are described in terms of $p$-subgroup invariants. The dimension can grow exponentially in $p$-rank or can track directly the order of maximal abelian $p$-subgroups in specific exceptional cases (e.g., $p=2$ or $p$ a Mersenne prime) (Robinson, 2020).

6. Quantum Deformations, PI Structure, and Growth Control

In quantum PI settings (e.g., multiparameter quantum Weyl algebras at roots of unity), the dimension of simple modules is bounded above by the PI degree, which is an explicit function of the orders of deformation parameters. All simple modules fall into finitely many torsion-type families, and the possible dimensions are explicit product-type polynomials in these orders, with the total degree bounded (for the second quantum Weyl algebra at roots of unity, at most cubic) (Bera, 2024). Thus, there is a complete absence of dimension-explosion: PI-theory and central structure rigidly constrain possible module sizes.

7. Geometric and Combinatorial Interpretations

Dimension growth often encodes geometric or combinatorial invariants. In Lie theory, dimension polynomials correspond to supports or multiplicities on Springer fibers, Kazhdan–Lusztig cells, or nilpotent orbits. In combinatorial settings, e.g., Boolean matrix algebras, the dimension formulas count lattice points or mappings hitting prescribed subsets, leading to inclusion–exclusion formulas. For quantum group modules, the GK dimension may coincide with combinatorial data such as the Coxeter length of a Weyl word, reflecting growth in both the classical and quantum settings (Chakraborty et al., 2017).

The study of dimension growth of simple modules thus serves as a bridge between noncommutative ring theory, geometric representation theory, and combinatorics, revealing both universal patterns and subtle case-dependent phenomena across algebraic frameworks.