Kadar–Yu Algebras: A Diagrammatic Framework

• Kadar–Yu algebras are a family of diagrammatic algebras that interpolate between Temperley–Lieb and Brauer algebras by imposing a left-height constraint on Brauer diagrams.
• They are generated by Temperley–Lieb and symmetric group elements, using diagrammatic multiplications and mixed relations to capture rich representation-theoretic structures.
• Their structure features iterated inflation, Chebyshev-type recurrence relations, and ties to Kazhdan–Lusztig theory and alcove geometry, unifying semisimple and non-semisimple phenomena.

The Kadar–Yu (KY) algebras form a distinguished tower of diagrammatic algebras interpolating between the Temperley–Lieb and Brauer algebras. Parameterized by a non-negative integer $\ell$ and a parameter $\alpha$ or $\delta$ from the base field $K$ (typically $K = \mathbb{C}$ or an algebraically closed field of characteristic zero), these algebras arise as endomorphism algebras in certain monoidal subcategories $J_\ell$ of the Brauer category. The structure and representation theory of KY algebras intertwine combinatorial, diagrammatic, and homological techniques, connecting directly to Kazhdan–Lusztig theory and alcove geometry, and generalizing classical semisimplicity paradigms through a family of generalized Chebyshev recurrences. Their study unifies a spectrum of algebraic and combinatorial objects, controlling both semisimple and non-semisimple phenomena.

1. Definition and Diagrammatic Presentation

Let $n, \ell \geq 0$ (or $\ell \geq -1$) and $K$ a unital commutative ring (often a field). The Brauer algebra $\operatorname{Br}_n(\delta)$, parameterized by $\delta \in K$, has as basis the set of $(n,n)$ Brauer diagrams: pairings of $2n$ points, with multiplication realized diagrammatically by vertical concatenation and with a factor of $\delta$ for each closed loop formed. The Kadar–Yu algebra $J_{\ell, n}(\delta)$ is defined as the span of those Brauer diagrams whose "left-height" does not exceed $\ell$, where left-height is the maximum, over all crossing points, of the minimal number of lines crossed from the face to the left boundary. This subspace is closed under multiplication, hence defines a subalgebra.

Algebraically, $J_{\ell,n}(\delta)$ is generated by:

• Temperley–Lieb generators $u_i$ $(1 \leq i < n)$ satisfying the standard Temperley–Lieb relations:
  • $u_i^2 = \delta u_i$,
  • $u_i u_{i\pm 1} u_i = u_i$,
  • $u_i u_j = u_j u_i$ for $|i-j|>1$.
• Symmetric group generators $s_m$ $(1 \leq m \leq \ell+1)$ satisfying Coxeter relations:
  • $s_m^2 = 1$,
  • $s_m s_{m+1} s_m = s_{m+1} s_m s_{m+1}$,
  • $s_m s_k = s_k s_m$ for $|m-k|>1$.
• Mixed relations reflecting that $s_m$ acts on the first $\min(\ell+2, n)$ propagating lines:
  • $s_m u_i = u_i s_m$ unless $|m-i| \leq 1$,
  • $s_m u_m s_m = u_{m+1}$ $(m=1,\dots,\ell)$,
  • $s_{\ell+1} u_i = u_i s_{\ell+1}$ for $i > \ell+1$.

Boundary cases: $J_{-1,n}(\delta) \cong \operatorname{TL}_n(\delta)$ is the Temperley–Lieb algebra, and $J_{n-2,n}(\delta) = \operatorname{Br}_n(\delta)$ the full Brauer algebra. These algebras thus interpolate between Temperley–Lieb and Brauer as $\ell$ varies.

2. Iterated Inflation Structure and Tower of Recollement

KY algebras naturally admit a tower structure as $\{J_{\ell,n}(\delta)\}_{n\geq 0}$. Each algebra admits a filtration

$$0 = I_{-2} \subset I_0 \subset I_2 \subset \dots \subset I_{n \text{ or } n-2}$$

where $I_m$ is the span of diagrams with at most $m$ propagating lines. The successive quotients satisfy

$$I_m / I_{m-2} \cong V_m \otimes_K K S_{m_\ell} \otimes_K V_m^*$$

with $m_\ell = \min(\ell+2, m)$ and $V_m$ the space of upper half-diagrams of appropriate left-height with $m$ propagating lines. This realizes $J_{\ell,n}(\delta)$ as an iterated inflation algebra, successively inflating symmetric group algebras $K S_{m_\ell}$ by these diagram modules.

This tower structure satisfies the six axioms (A1–A6) of Cox–Martin–Parker–Xi (CMPX), defining a "tower of recollement" (ToR):

• (A1) Localisation by idempotents reduces $J_{\ell,n}$ to $J_{\ell,n-2}$.
• (A2) Quasi-heredity with explicit heredity chains.
• (A3) Embedding into algebras of higher $n$ by adding strands.
• (A4) Bimodule isomorphisms relating $J_{\ell,n}$ via idempotents to bimodules over $J_{\ell,n-1}, J_{\ell,n-2}$.
• (A5) Restriction of cell modules admits a filtration supported on $(p\pm1,?)$.
• (A6) Each simple appears as a subfactor of a restriction from higher rank.

This realizes KY algebras as a ToR in the sense of [Cox–Martin–Parker–Xi, 2006] and controls the homological and cellular structure of representations (Alraddadi et al., 2024).

3. Standard Modules, Gram Determinants, and the Chebyshev Paradigm

Standard (cell) modules $\Delta^{n}_{(p,\lambda)}$ of $J_{\ell,n}$ are indexed by pairs $(p,\lambda)$ where $0 \leq p \leq n$, $n-p$ even, and $\lambda \vdash \min(p, \ell+2)$. The construction involves a primitive idempotent in $J_{\ell,p}$ and the diagrammatic module $J_\ell(n,p)$, modulo diagrams with $<p$ propagating lines. For $p = n$, the head standard module $\Delta^{n}_{(n,\lambda)}$ coincides with the Specht module $S_\lambda$.

Cell modules admit a $*$-contravariant bilinear form (where $*$ reverses diagrams). The Gram determinant $\|\Delta^{n}_{(p,\lambda)}\|$ can be computed in terms of $\delta$ or $\alpha$, and for the important family $p=n-2$, these determinants satisfy a Chebyshev recurrence:

$$P_{n+1}(\alpha) = \alpha P_n(\alpha) + P_{n-1}(\alpha)$$

Each partition $\lambda \vdash \ell+2$ yields a monic polynomial $C^{(\lambda)}(\alpha)$ and reduced Chebyshev-type series $P_n^{(\lambda)}(\alpha)$, such that for $n \geq \ell+4$,

$$\|\Delta^{n}_{(n-2, \lambda)}\| = C^{(\lambda)}(\alpha)[P_n^{(\lambda)}(\alpha)]^{\dim S_\lambda}.$$

For example, for $\ell=-1$ (Temperley–Lieb), $P_n^{()} = U_{n-1}(\alpha/2)$ recovers the Chebyshev $U$-polynomials, while for large $\ell$ (Brauer), the determinants correspond to hook-length products (Morris et al., 31 Dec 2025).

Low-rank cases demonstrate the roles of partitions and the combinatorics of standard modules:

• $\ell=0$: $\lambda \in \{(2), (1^2)\}$, each with explicit $C^{(\lambda)}$ and Chebyshev-series $P_n^{(\lambda)}$.
• $\ell=1$: $\lambda$ runs over all partitions of $3$; similar explicit determinants arise.

4. Semisimplicity and Non-Semisimple Representation Theory

The semisimplicity of $J_{\ell,n}(\delta)$ over $\mathbb{C}$ is controlled by the Gram determinants. If $\delta \notin \mathbb{R}$, all $\|\Delta^{n}_{(p,\lambda)}\| \neq 0$ for all $p,\lambda$, so the algebra is semisimple. In this regime, every cell module is simple, labeled by $(p,\lambda)$ as above, and the regular module decomposes accordingly:

$$J_{\ell,n}(\delta) \cong \bigoplus_{p,\lambda} \operatorname{End}_K(\Delta(p,\lambda))$$

(Alraddadi et al., 2024).

For real $\delta$ or specializations of $\alpha$ (e.g., roots of the associated Chebyshev series), the algebras may fail to be semisimple. At each vanishing determinant (a root $\alpha_0$ of some $P_n^{(\lambda)}$), there exists a non-split map between standard modules, controlled by explicit "bootstrap" elements $\xi^{(n)}$. The structure of non-semisimple blocks, submodules, composition series, and socles is deterministic, exhibiting direct links to alcove geometry and representation-theoretic walls.

The general pattern in the non-semisimple case is controlled by the interplay between the tower functors (localisation and globalisation in the ToR) and the stratification of the module category via the Chebyshev determinants (Morris et al., 31 Dec 2025).

5. Alcove Geometry, Combinatorics, and the Bratteli/Rollet Graphs

KY algebras' module categories demonstrate a pronounced alcove-geometric structure. The indexing set $(p,\lambda)$ organizes into the Rollet (projected Bratteli) graph, projecting the ordinary Bratteli diagram by omitting certain diagrammatic features ("$e$-strands"). The head comprises partitions of size at most $\ell+2$, from which arms emanate corresponding to larger $p$.

This labeling system can be embedded into $\mathbb{R}^r$ as the $1$-skeleton of a finite or affine Weyl alcove tessellation. For $\ell=-1$ (Temperley–Lieb), the set is one-dimensional and semisimplicity corresponds to the non-vanishing of a parameter outside the quantum-group root-of-unity locus $2 \cos(k\pi/(n+1))$. For higher $\ell$, the complexity escalates to higher-rank affine Weyl group reflection hyperplanes.

Non-semisimplicity loci (roots of Chebyshev series) correspond to reflection hyperplanes in $\alpha$-space; the combinatorics of add/remove box moves on partitions (Young lattice) realize the percolation of submodules and the precise positioning of module filtrations. Exact sequences and composition factor structures are thus articulated geometrically (Morris et al., 31 Dec 2025).

6. Connections, Physical Motivation, and Low-Rank Examples

Kadar–Yu algebras are physically motivated as deformations and extensions of symmetric group and Temperley–Lieb representation theory in the setting of statistical mechanics, integrable models, and categorical constructions. They interpolate between Catalan (Temperley–Lieb) and double-factorial (Brauer) combinatorics, further connecting to Kazhdan–Lusztig theory via the interplay between cellular, quasi-hereditary, and topos-theoretic frameworks.

Low-rank instances explicitly demonstrate how the intermediate algebras possess richer representation theory than either extreme:

• For $n=4$, $\ell=0$, simples are
  • $\Delta(4, \Box \vdash 2)$ (dim $2$),
  • $\Delta(2, \Box \vdash 2)$ (dim $4$),
  • $\Delta(0, \emptyset)$ (dim $3$),
  • giving $$\operatorname{Br}_4(\delta) \cong M_2(\mathbb{C}) \oplus M_4(\mathbb{C}) \oplus M_3(\mathbb{C})$$ for non-real $\delta$.

The passage from $\ell=-1$ to $\ell=n-2$ recovers the interpolation between Temperley–Lieb and Brauer from both diagrammatic and representation-theoretic standpoints, always controlled by the relevant determinant formulas and alcove geometry.

7. Summary Table: Special Cases

$\ell$	Algebra	Description	Semisimplicity Condition
$-1$	Temperley–Lieb	No $s_j$ generators	$\delta \notin \{2\cos(\frac{\pi}{k})\}$
$0$	"Rook–Brauer"/boundary-TL	First additional crossings	Non-real $\delta$; $P_n^{(\lambda)}(\delta) \neq 0$
$n-2$	Brauer	Full diagram algebra	Non-real $\delta$

Failures of semisimplicity correspond to roots of specialized Chebyshev polynomials $P_n^{(\lambda)}(\alpha) = 0$ and yield non-trivial homological and combinatorial phenomena.

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Kadar–Yu algebras thus constitute a unifying and tractable family of diagrammatic algebras, whose module-theoretic and combinatorial properties capture a broad interpolation between well-understood classical examples, regulated by generalizations of the Chebyshev recurrence and alcove geometric framework (Alraddadi et al., 2024, Morris et al., 31 Dec 2025).