Left Cell Invariance

• Left Cell Invariance is a property in representation theory and algebraic combinatorics where left cell structures remain unchanged under functors, automorphisms, and perturbations.
• It underpins the analysis in affine Weyl groups and Hecke algebras by preserving combinatorial and categorical structures through defined descent sets and cellular maps.
• The concept has practical applications across spectral analysis, cellular automata, and 2-representation theory, ensuring stability and invariant behavior in complex mathematical systems.

Left cell invariance is a foundational concept in representation theory, algebraic combinatorics, operator theory, and mathematical physics. It encodes the phenomenon where structures, representations, or invariants associated to "left cells"—combinatorial or categorical subsets defined via preorder or equivalence relations—remain unchanged under the action of certain functors, automorphisms, translations, or perturbations. The precise content and consequences of left cell invariance depend on context: in affine Weyl groups, Hecke algebras, cellular automata, 2-representation theory, image registration, and operator spectra, distinct but structurally related invariance theorems and constructions manifest the concept.

1. Canonical Left Cells in Coxeter and Affine Weyl Groups

In the setting of affine Weyl groups, left cells are defined combinatorially via the Kazhdan–Lusztig theory. Let $W=W_0 \ltimes P$ (extended affine Weyl group), with $W_0$ the finite Weyl group and $P$ the weight lattice. The left and right descent sets $L(w),\ R(w)$ for $w\in W$ identify structure-preserving subsets. A canonical left cell is characterized by

$Y_0 = \{\,w\in W\mid R(w)\subseteq\{s_0\}\,\},$

where $s_0$ is the affine simple reflection (Xi, 2011). For a two-sided cell $c$, $Y_0\cap c$ is a left cell, yielding a canonical representative. Canonical left cells admit refined formulations involving anti-dominant weights ($P^-$) and descent set inclusions:

$$Y_0 = \{\,w x\mid w\in W_0,\ x\in P^-,\ R(w)\subseteq L(x)\,\}.$$

This combinatorial rigidity is central to left cell invariance: modules, representations, or ideals built from canonical left cells in the lowest two-sided cell $c_0$ retain structural invariants under algebra automorphisms and functors.

2. Left Cell Invariance in Hecke Algebras and Vogan Invariants

Kazhdan–Lusztig cells partition elements of Coxeter groups and control the structure of Hecke algebra modules. Vogan's left cell invariants generalize this by inductively refining invariants with combinatorial operators. The base step equates right descent sets:

$x\approx_0 y\iff R(x)=R(y).$

Inductive steps use "neighbor" maps on dihedral subgroups, with recursive refinement:

$$x\approx_n y\iff x\approx_{n-1} y\;\text{and for all relevant st-pairs,}\;\text{the neighbor sets are matched recursively}.$$

A left cellular map $\delta\colon W\to W$ is "cellular" if it preserves left cells and induces module isomorphisms (Bonnafé et al., 2015). The framework extends to Hecke algebras with unequal parameters, maintaining invariance under left cellular operations—even as parameters vary across conjugacy classes. Left cell invariance here is both combinatorial (invariance under cellular maps and induced permutations) and categorical (the $\tau$-invariant fully determines the left cell in symmetric groups).

3. Left Cell Invariance in Cellular Automata and Group Structure

In linear synchronous cellular automata (CA), left and right shift automata act invertibly on global configurations:

$$TL_m(C)(i) = C(i+m),\quad TR_m(C)(i)=C(i-m),\quad i\in\mathbb{Z}.$$

Composing left and right shifts yields the identity, and the collection of all such shift automata plus the identity forms an abelian group under composition (Ghosh et al., 2017):

$(TL_m \circ TR_n)(C)(i) = C(i + (m-n)).$

Left cell invariance in CA denotes that local cell patterns are transported undistorted under shifts. The invariance of local structure (neighborhood relationships) is maintained under group action—mirroring combinatorial invariance in algebraic contexts.

4. Left Cell Invariance in 2-Representation Theory and Differential Graded Categories

In 2-category settings, particularly for projective functors over star algebras, left cell invariance controls the classification of simple transitive 2-representations. For Dynkin type $A_2$, all simple transitive 2-representations of the corresponding left cell subcategory are equivalent to cell 2-representations (Zimmermann, 2018). In larger star algebras, matrix analysis predicts many simple transitive 2-representations that are not cell 2-representations, but their classification still hinges on left cell combinatorics.

In differential graded (dg) 2-categories, left cells are defined via strong and weak partial preorders:

• Strong preorder $F \leq_L G$: $G$ is in the thick closure generated by $F$.
• Weak preorder $F \preceq_L G$: relates to dg ideals of identity morphisms (Laugwitz et al., 2022).

A key result is a bijection between maximal ideal spectra of strong and weak cell 2-representations, establishing dg-equivalence up to quotient by maximal ideals. Thus, cell 2-representations built from left cells are invariants of the cell structure, unaffected by the choice of strong/weak preorder.

5. Left Cell Invariance in Affine Weyl Groups: Translations and Cell Structure

Translations in affine Weyl groups exhibit "cell invariance" under conjugation and powers. For $x$ a translation and $w\in W_0$ in the extended group, the following holds:

$$x \sim_{LR} w x w^{-1},\quad x \sim_L x^m,\quad x \sim_R x^m\qquad \forall m>0.$$

This invariance controls the organization of left and right cells in the second lowest two-sided cell, $\Omega_{qr}$ (Qiu, 2020). The conjecture and its refinement (proved for affine types $\tilde{A}_{n-1}$, $\tilde{G}_2$) establish that dominant translations with one coordinate zero (excluding the lowest two-sided cell) parametrize distinct right cells, with the number of left cells in $\Omega_{qr}$ equal to $|W_0|/2$.

6. Left Cell Invariance in Operator Theory and Perturbations

Analyticity and spectral invariance under rank-one perturbations exemplify operator-theoretic manifestations of left cell invariance (Chavan et al., 2022). For the multiplication operator $\mathscr{M}_z$ and a perturbation $f\otimes g$ (with $g\in\ker T^*$), the left spectrum remains unchanged except at possibly one point:

$$\sigma_l(T + f \otimes g) \setminus \{ \langle f, g \rangle \} = \sigma_l(T) \setminus \{ \langle f, g \rangle \}.$$

Here, $\langle f, g \rangle$ may be a simple eigenvalue if it appears; otherwise, the left spectrum displays invariance under the perturbation.

7. Left Cell Invariance in Algebraic Combinatorics: KL Preorder and Tableaux Dominance

In the symmetric group $S_n$, Kazhdan–Lusztig left cell preorders reflect combinatorial invariance via Robinson–Schensted–Knuth (RSK) correspondence. For $x \leq_L y$,

$Q(y) \unrhd Q(x),$

where $Q(w)$ denotes the recording tableau and $\unrhd$ is dominance order on tableaux (He et al., 2021). KL basis elements expand into Murphy basis elements, mirroring dominance:

$$C'_w = v^{-\ell(w_x,0)} m_{u_w, d_w} + \sum_{(s,t) \succ (u_w, d_w)} r_{s,t} m_{s,t}.$$

Left cell invariance here guarantees that the cell structure is preserved—combinatorially parameterized by tableaux dominance, and structurally visible in the triangularity of basis transitions.

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Left cell invariance thus synthesizes combinatorial rigidity, module-theoretic functoriality, spectral stability, and categorical invariance across diverse mathematical domains. The central theme is that the structures induced by left cells (whether as subsets, modules, representations, or categorical objects) possess well-defined invariants under actions, automorphisms, and perturbations prescribed by the underlying theory, thereby anchoring significant advances in modern representation theory, algebraic combinatorics, operator theory, and categorical algebra.