Root-of-unity restricted Q-system conjecture

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Fix a level ℓ\ell and let Qm(a)=dim⁡qres Wm(a)Q^{(a)}_m=\dim_q\mathrm{res}\,W^{(a)}_m be the root-of-unity quantum dimensions of the restricted Kirillov–Reshetikhin modules, with a∈Ia\in I and ℓa\ell_a the corresponding restricted boundary. Restricted Q-system conjecture. The quantities Qm(a)Q^{(a)}_m satisfy the level-ℓ\ell restricted Q-system and, more strongly, for every a∈Ia\in I,

Qm(a)=Qℓa−m(a)(0≤m≤ℓa),Q^{(a)}_m=Q^{(a)}_{\ell_a-m}\qquad(0\le m\le\ell_a), Qm(a)<Qm+1(a)(0≤m<[ℓa/2]),Q^{(a)}_m<Q^{(a)}_{m+1}\qquad\left(0\le m<\left[\ell_a/2\right]\right),

and

Qℓa+j(a)=0(1≤j≤tah∨−1).Q^{(a)}_{\ell_a+j}=0\qquad(1\le j\le t_a h^\vee-1).

Here [ℓa/2][\ell_a/2] is the greatest integer not exceeding ℓa/2\ell_a/2. The claim is a proposed level truncation of the unrestricted Q-system at a root of unity; the supplied text gives no resolution.

References

Primary source

Atsuo Kuniba, Tomoki Nakanishi and Junji Suzuki, “T-systems and Y-systems in integrable systems”, arXiv:1010.1344 (2014).

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