Arm property and closed forms for standard-module Gram determinants of the Kadar–Yu algebras (Conjectures 3.27.1, 3.31.1, 3.32.1 and 5.2.1 of Morris–Martin)

The Kadar–Yu algebras $J_{l,n}(\alpha)$ form a one-parameter family of diagram-algebra towers, indexed by an integer $l\ge-1$, that interpolate between the Temperley–Lieb algebras ($l=-1$) and the Brauer algebras ($l\to\infty$). Morris and Martin (2025) reduced the Gram determinants of the one-cup ('bootstrap') standard modules of every tower $J_l$ to families of Chebyshev-type polynomials $P_n^{(\lambda)}$ and conjectured, but did not prove, that these closed forms extend to all heights and all partition labels, and that a 'marginal vertex identity' (the arm property) reconstructs every remaining standard-module Gram determinant from the bootstrap data. The problem posed here is to prove or refute these four conjectures; an affirmative resolution determines every standard-module Gram determinant in every Kadar–Yu tower, and hence the complete non-semisimple parameter locus of these algebras.

Background

The Temperley–Lieb algebra is the span of planar matchings of $2n$ points by non-crossing arcs ('cups and caps' with propagating lines through the middle), and the Brauer algebra is its enlargement allowing crossings; both arise as transfer-matrix algebras in statistical mechanics and their complex representation theory — semisimple and non-semisimple — is classical, with the non-semisimple theory controlled by roots of unity (Temperley–Lieb) and by an integral paradigm connected to Kazhdan–Lusztig theory (Brauer). Kadar, Martin and Yu (2014) introduced, for each integer $l\ge-1$, a tower of subalgebras $J_{l,n}(\alpha)$ of the Brauer algebra — the Kadar–Yu (or KMY) algebras, with complex parameter $\alpha$ — where $l$ bounds the 'left height' of allowed crossings, so that $J_{-1,n}$ is Temperley–Lieb and the towers run toward the Brauer algebra as $l$ grows. Their semisimple theory is understood; Alraddadi and Parker (2024) showed the $J_{l,n}(\delta)$ are iterated inflation algebras satisfying the Cox–Martin–Parker–Xi 'towers of recolment' (ToR) axioms (a framework relating the module categories of the $J_{l,n}$ by inclusion and by idempotent localisation, i.e. by Schur-functor-style functors $eJ_{l,n+2}e\cong J_{l,n}$), and proved semisimplicity over $\mathbb{C}$ whenever the parameter is non-real. The non-semisimple theory for real parameters has remained largely open.

A standard tool for such diagram algebras is the Gram matrix of the contravariant bilinear form on a standard module $\Delta^{(n)}_{(p,\lambda)}$ (the span of diagrams with exactly $p$ propagating lines and a symmetrised group action labelled by a partition $\lambda$): the algebra is semisimple at parameter $\alpha$ only if none of these Gram determinants $D^{(n)}_{(p,\lambda)}(\alpha)$ vanishes, and vanishing determinants flag nontrivial homomorphisms between standard modules. Morris and Martin (2025) proved (their Theorem 3.19) that for fixed $l$ and each $\lambda\vdash l+2$ there are a polynomial $C^{(\lambda)}$ and 'ramping' polynomial series $P_n^{(\lambda)}$ (a ramping series is one that is eventually monic with $\deg P_{n+1}=\deg P_n+1$; here it is determined by Chebyshev-type three-term recurrences — Chebyshev polynomials being the classical polynomials $T_k$, $U_k$ with trigonometric root structure) such that, for $n\ge l+4$ and up to an overall complex scalar, every one-cup ('bootstrap', i.e. $m=1$, so $n=p+2$) determinant factors as $D^{(n)}_{(p,\lambda)}=C^{(\lambda)}\,(P^{(\lambda)}_n)^{d_\lambda}$ with $d_\lambda=\dim S^\lambda$, the dimension of the Specht module of the symmetric group labelled by $\lambda$. To reach all standard modules they organise the index set $(p,\lambda)$, $p$ propagating lines, into the Rollet graph $R_l$ (the projection of the Bratteli diagram of restriction of standard modules along the tower, with an edge $(p,\lambda)-(p+1,\mu)$ when $\mu$ is obtained from $\lambda$ by adding a box or $\mu=\lambda$; the head of $R_l$ is its initial subgraph agreeing with the Young graph of partitions under box-addition, the arms are the remaining half-lines — one $\lambda$-arm for each $\lambda\vdash l+2$ — and the shoulders are the last head vertices, where the arms begin), and define the marginal vertex function

$$\mathcal{V}^{n}_{(p,\lambda)}:=\frac{D^{(n=p+2m)}_{(p,\lambda)}}{\prod_{v\ \text{adjacent to}\ (p,\lambda)\ \text{in}\ R_l}D^{(n-1)}_{(v)}} .$$

Since the determinants are recovered from the $\mathcal{V}$ by recursion, a closed form for $\mathcal{V}$ would determine all standard determinants. Morris–Martin's Theorem 5.2 proves the identity $\mathcal{V}^{n}_{(p,\lambda)}=\mathcal{C}^{n}_{(p,\lambda)}$ — where $\mathcal{C}^{n}_{(p,\lambda)}$ is the explicit product over arm vertices $(p+1,\mu)$ adjacent to $(p,\lambda)$ of the ratios $(P^{(\mu)}_n/P^{(\mu)}_{n-1})^{\dim\Delta^{n-1}_{(p+1,\mu)}}$ — only in restricted ranges: $m=1$ for all $l$ and $p\ge l+3$; all $n$ when $l=-1$ (the Temperley–Lieb case, where it recovers the classical root-of-unity determinant theory); and small finite ranges for $l=0,1$. Their Conjecture 5.2.1 asserts the identity for all $l\ge-1$, $p\ge l+2$, $m\ge1$. The needed closed forms for the polynomials $P^{(\lambda)}_n$ themselves are established only for special partition shapes and bounded $l$ (verified up to $l\le 12$ by machine), and are conjectured in general (their Conjectures 3.27.1, 3.31.1, 3.32.1, the last asserting integrality of all coefficients and integrality of the roots of $C^{(\lambda)}$). The four conjectures remain open in the cited literature.

Problem Statement

Prove or refute the following four conjectures of Morris–Martin (arXiv:2512.24535), which together determine every standard-module Gram determinant of every Kadar–Yu tower $J_l$.

(i) Conjecture 5.2.1 (the arm property): for every integer $l\ge-1$, every $p\ge l+2$, every $m\ge1$, and every partition $\lambda\vdash l+2$, the marginal vertex function satisfies

$$\mathcal{V}^{n}_{(p,\lambda)}=\mathcal{C}^{n}_{(p,\lambda)},\qquad n=p+2m,$$

where $\mathcal{V}^{n}_{(p,\lambda)}=D^{(n)}_{(p,\lambda)}\big/\prod_{v\sim(p,\lambda)}D^{(n-1)}_{(v)}$ (product over vertices adjacent to $(p,\lambda)$ in the Rollet graph $R_l$), and $\mathcal{C}^{n}_{(p,\lambda)}$ is the explicit product, over arm vertices $(p+1,\mu)$ adjacent to $(p,\lambda)$ in $R_l$, of the factors $(P^{(\mu)}_n/P^{(\mu)}_{n-1})^{\dim\Delta^{n-1}_{(p+1,\mu)}}$, with $P^{(\mu)}_k$ the ramping polynomials of Theorem 3.19. Since the Gram determinants are recovered from the $\mathcal{V}$ by recursion, and the finitely many head and shoulder determinants are directly computable, an affirmative answer determines every standard-module Gram determinant $D^{(n)}_{(p,\lambda)}(\alpha)$ of $J_{l,n}(\alpha)$, and hence the full non-semisimple parameter locus of each tower.

(ii) Conjecture 3.27.1: the explicit closed-form expressions (3.3.1)–(3.3.4) of the source for the polynomial series $P^{(l'+2)}_{l'+4+k}$ and $P^{(1^{l+2})}_{l+4+k}$ (one-row and one-column partition labels), stated as combinations of Chebyshev-type polynomials $PU_k(\alpha)$ with $l$-dependent coefficients ($PU_k$ being the source's normalisation of the Chebyshev polynomials of the second kind, $PU_k(x)=U_{k-1}(x/2)$), hold for all $l\ge-1$ and all $l'\ge0$ (the source verifies them for $-1\le l\le12$, $0\le l'\le12$).

(iii) Conjecture 3.31.1: the explicit closed-form expressions (3.3.7)–(3.3.10) of the source — closed forms for the polynomials $C^{(l'+1,1)}$ and $C^{(l+1,1)^T}$ and the ramping series $P^{(l''+1,1)}_k$ and $P^{(l+1,1)^T}_k$ attached to the hook partition $(l'+1,1)$ (a two-part partition: one row plus one box) and its transpose — hold for all $l\ge1$, $l'\ge4$ and $l''\ge2$.

(iv) Conjecture 3.32.1: for all $l\ge-1$ and all $\lambda\vdash l+2$, the polynomials $C^{(\lambda)}$ and $P^{(\lambda)}_n$ of Theorem 3.19 have integer coefficients, and $C^{(\lambda)}$ has only integer roots (the source proves this for $l\le2$, for $\lambda=(l+2),(1^{l+2})$ with $l\le12$, and for $\lambda=(l+1,1),(l+1,1)^{T}$ with $l\le7$).

Each of (i)–(iv) is a universal statement over the full indicated ranges, with no additional restrictions on $l$, $p$, $m$, or the partition $\lambda$; the precise right-hand sides of the closed forms in (ii) and (iii) are those printed in Section 3.3 of the source and must be consulted there verbatim.

References

1. Benjamin Morris, Paul P. Martin, On semisimplicity criteria and non-semisimple representation theory for the Kadar-Yu algebras, 2025. DOI 10.48550/arxiv.2512.24535, https://arxiv.org/abs/2512.24535
2. Zoltan Kadar, Paul P. Martin, Shona Yu, On geometrically defined extensions of the Temperley-Lieb category in the Brauer category, 2014. DOI 10.48550/arxiv.1401.1774, https://arxiv.org/abs/1401.1774; journal version: Mathematische Zeitschrift 293 (2019) 1247–1276, DOI 10.1007/s00209-019-02246-4
3. Nouf Alraddadi, Alison Parker, On semi-simplicity of KMY algebras, 2024. DOI 10.48550/arxiv.2407.07028, https://arxiv.org/abs/2407.07028
4. Anton Cox, Paul Martin, Alison Parker, Changchang Xi, Representation theory of towers of recolment: theory, notes, and examples, 2004. arXiv math/0411395, https://arxiv.org/abs/math/0411395; journal version: Journal of Algebra 302 (2006) 340–360, DOI 10.1016/j.jalgebra.2006.01.009
5. Zoltan Kadar, Paul P. Martin, Geometric partition categories: On short Brauer algebras and their blob subalgebras, 2020. DOI 10.48550/arxiv.2002.05478, https://arxiv.org/abs/2002.05478