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Quantization of Virtual Grothendieck Rings and Their Structure Including Quantum Cluster Algebras

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Abstract

The quantum Grothendieck ring of a certain category of finite-dimensional modules over a quantum loop algebra associated with a complex finite-dimensional simple Lie algebra \(\mathfrak {g}\) has a quantum cluster algebra structure of skew-symmetric type. Partly motivated by a search of a ring corresponding to a quantum cluster algebra of skew-symmetrizable type, the quantum virtual Grothendieck ring, denoted by \(\mathfrak {K}_q(\mathfrak {g})\), is recently introduced by Kashiwara and Oh (Math Z 303(2):42, 2023) as a subring of the quantum torus based on the (q, t)-Cartan matrix specialized at \(q=1\). In this paper, we prove that \(\mathfrak {K}_q(\mathfrak {g})\) indeed has a quantum cluster algebra structure of skew-symmetrizable type. This task essentially involves constructing distinguished bases of \(\mathfrak {K}_q(\mathfrak {g})\) that will be used to make cluster variables and generalizing the quantum T-system associated with Kirillov–Reshetikhin modules to establish a quantum exchange relation of cluster variables. Furthermore, these distinguished bases naturally fit into the paradigm of Kazhdan–Lusztig theory and our study of these bases leads to some conjectures on quantum positivity and q-commutativity.

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Notes

  1. In the main body of this paper, we sometimes call it t-character by replacing the role of q by t.

  2. Our convention is a variation of the Coxeter–Dynkin diagram in the sense that we connect vertices with single edges only. See the examples for the finite types. We will call them Dynkin diagrams for simplicity.

  3. In [47, 48], \(\underline{\widetilde{\textsf{B}}}\) and \(\widetilde{\textsf{b}}\) are used instead of \(\widetilde{\underline{\textsf{R}}}\) and \(\widetilde{\textsf{r}}\), respectively.

  4. Visually, they are slant.

  5. When we replace valued arrows with usual arrows, it is the usual repetition quiver \(\widehat{\triangle }\) (see [47] for non-simply-laced types).

  6. It is usually called the q-character homomorphism in the literature.

  7. In our introduction, we use \(\overline{\mathcal {K}}_{1,\texttt{t},d}(\mathfrak {g})\) instead.

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Acknowledgements

The authors would like to thank the referees for reading our manuscript carefully and making so many constructive comments. I.-S. Jang was supported by Incheon National University Research Grant in 2023 (No. 2023-0205), K.-H. Lee was partially supported by a grant from the Simons Foundation (#712100), and S.-j. Oh was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2022R1A2C1004045).

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Authors and Affiliations

  1. Department of Mathematics, Incheon National University, Incheon, 22012, South Korea

    Il-Seung Jang

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Kyu-Hwan Lee

  3. Department of Mathematics, Sungkyunkwan University, Suwon, South Korea

    Se-** Oh

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Examples for (Quantum) Positivity

Examples for (Quantum) Positivity

1.1 Quantum positivity and speciality of KR-polynomials

In this subsection, we provide examples for Conjecture 1 and Conjecture 2. Recall that

$$\begin{aligned} F_q(\underline{X_{i,p}}) = E_q(\underline{X_{i,p}}) = L_q(\underline{X_{i,p}}), \end{aligned}$$

and the quantum positivity of \(F_q(\underline{X_{i,p}})\) for types \(B_3\) and \(G_2\) are already verified (up to shift of spectral parameters) in Example 5.16, Example 5.24 (for type \(G_2\)), and Example 5.25 (for type \(B_3\)). In what follows, we provide the formulas for fundamental polynomials for type \(C_3\). Those elements may be obtained from the q-algorithm (cf. Example 5.16) or the quantum cluster algebra algorithm in Proposition 8.6, so we skip the details. The explicit formulas of \(F_q(\underline{X_{i,0}})\)’s for type \(C_3\) are given as follows (under the same convention in the previous examples):

figure d

Since \(F_q(\underline{X_{i,p}}) = \textsf{T}_p( F_q(\underline{X_{i,0}}))\), we verify the quantum positivity of all fundamental polynomials for type \(C_3\). We further remark that the quantum positivity of \(F_q(\underline{X_{i,p}})\) for type \(F_4\) also holds (with the help of computer program).

For type \(G_2\), one may compute

$$\begin{aligned} \begin{aligned}&L_q(\underline{X_{1,6}^3}) = E_q(\underline{X_{1,6}^3}) = F_q(\underline{X_{1,6}^3}), \quad L_q(\underline{X_{2,5}X_{2,7}}) = F_q(\underline{X_{2,5}X_{2,7}}), \\&E_q(\underline{X_{2,5} X_{2,7}}) = L_q(\underline{X_{2,5} X_{2,7}}) + P_{X_{2,5} X_{2,7}, X_{1,6}^3}(q) L_q (\underline{X_{1,6}^3}), \end{aligned} \end{aligned}$$
(A.2)

where \(P_{X_{2,5} X_{2,7}, X_{1,6}^3}(q) = q^3 \in q\mathbb {Z}_{\geqslant 0}[q]\). Then the quantum positivity of \(L_q(\underline{X_{1,6}^3})\) follows from Example 5.24 and the definition of \(E_q(\underline{X_{1,6}^3})\). Moreover, it follows from (A.2), Example 5.16, and Example 5.24 that the quantum positivity of \(L_q(\underline{X_{2,5} X_{2,7}})\) also holds.

In general, for \(\underline{m}^{(i)}[p,s]\) with \(|p-s|\leqslant 2\), the computation of \(L_q(\underline{m}^{(i)}[p,s])\) is similar with the one for (A.2), since \(E_q(\underline{m}^{(i)}[p,s])\) has only two dominant monomials thanks to Theorem 6.9. It follows from Proposition 5.23 that this implies that \(L_q(\underline{m}^{(i)}[p,s]) = F_q(\underline{m}^{(i)}[p,s])\). For example, when \(\textsf{g}\) is of type \(B_3\),

$$\begin{aligned} \begin{aligned}&E_q(\underline{X_{1,0} X_{1,2}}) = q F_q(\underline{X_{1,0}}) * F_q(\underline{X_{1,2}}) =L_q(\underline{X_{1,0} X_{1,2}}) + q^2 L_q (\underline{X_{2,1}}), \\&E_q(\underline{X_{2,0} X_{2,2}}) = q^{-1} F_q(\underline{X_{2,0}}) * F_q(\underline{X_{2,2}}) = L_q(\underline{X_{2,0} X_{2,2}}) + q^2 L_q(\underline{X_{1,1}X_{3,1}^2}),\\&E_q(\underline{X_{3,0} X_{3,2}}) = F_q(\underline{X_{3,0}}) * F_q(\underline{X_{3,2}}) = L_q(\underline{X_{3,0} X_{3,2}}) + q L_q(\underline{X_{2,1}}). \end{aligned} \end{aligned}$$
(A.3)

Here \(L_q(\underline{X_{1,1}X_{3,1}^2})=E_q(\underline{X_{1,1}X_{3,1}^2})=F_q(\underline{X_{1,1}X_{3,1}^2})\). Furthermore, one may check that the quantum positivity of \(L_q(\underline{X_{i,0}X_{i,2}})\) holds from Example 5.25 and (A.3). Similarly, one may have an analog of (A.3) for type \(C_3\) with (A.1), which implies the quantum positivity of \(L_q(\underline{X_{i,0}X_{i,2}})\) in this case. However, we cannot use the same argument in general because it does not seem to be easily determined by direct computation how many dominant monomials \(E_q(\underline{m}^{(i)}[p,s])\) have for higher levels.

1.2 Quantum positivity of non KR-polynomials

In this subsection, we observe some examples in which \(L_q(\underline{m})\) has the quantum positivity for a dominant monomial m different from the KR-monomials.

Example A.1

Let us consider the case of type \(C_2\). Then the fundamental polynomials \(F_q(\underline{X_{1,2}})\) and \(F_q(\underline{X_{2,5}})\) are given as follows:

$$\begin{aligned} \begin{aligned}&F_q(\underline{X_{1,2}}) = q^{\frac{1}{2}} \widetilde{X}_{1,2} + q^{\frac{3}{2}} \widetilde{X}_{2,3}*\widetilde{X}_{1,4}^{-1} + q^{\frac{1}{2}} \widetilde{X}_{1,4}*\widetilde{X}_{2,5}^{-1} + q^{-\frac{1}{2}}\widetilde{X}_{1,6}^{-1}, \\&F_q(\underline{X_{2,5}}) = q \widetilde{X}_{2,5} + q^{2} \widetilde{X}_{1,6}^{2}*\widetilde{X}_{2,7}^{-1} + (q^{-1} + q) \widetilde{X}_{1,6}\widetilde{X}_{1,8}^{-1} + q^{2} \widetilde{X}_{2,7}*\widetilde{X}_{1,8}^{-2} + q^{-1} \widetilde{X}_{2,9}^{-1}. \end{aligned} \end{aligned}$$
(A.4)

It follows from (A.4) that

$$\begin{aligned} E_q(\underline{X_{1,2} X_{2,5}}) = q F_q(\underline{X_{1,2}}) * F_q(\underline{X_{2,5}}) = L_q(\underline{X_{1,2} X_{2,5}}) + q^2 L_q(\underline{X_{1,4}}), \end{aligned}$$
(A.5)

where \(L_q(\underline{X_{1,2} X_{2,5}}) = F_q(\underline{X_{1,2} X_{2,5}})\) and \(P_{X_{1,2}X_{2,5},\,X_{1,4}}(q) = q^2 \in q\mathbb {Z}[q]\). Then the quantum positivity of \(L_q(\underline{X_{1,2} X_{2,5}})\) follows from the formula (that may be computed with (A.4) and (A.5)) as shown below:

Example A.2

Let us consider the case of type \(B_2\). Then the fundamental polynomials \(F_q(\underline{X_{1,2}})\) and \(F_q(\underline{X_{2,5}})\) are given as follows (cf. (A.4)):

$$\begin{aligned} \begin{aligned} F_q(\underline{X_{1,2}}) =\,&q \widetilde{X}_{1,2} + q^{2} \widetilde{X}_{2,3}^{2}*\widetilde{X}_{1,4}^{-1} + (q^{-1} + q) \widetilde{X}_{2,3}*\widetilde{X}_{2,5}^{-1} + q^{2} \widetilde{X}_{1,4}*\widetilde{X}_{2,5}^{-2} + q^{-1} \widetilde{X}_{1,6}^{-1}, \\ F_q(\underline{X_{2,5}}) =\,&q^{\frac{1}{2}} \widetilde{X}_{2,5} + q^{\frac{3}{2}} \widetilde{X}_{1,6}*\widetilde{X}_{2,7}^{-1} + q^{\frac{1}{2}} \widetilde{X}_{2,7}*\widetilde{X}_{1,8}^{-1} + q^{-\frac{1}{2}} \widetilde{X}_{2,9}^{-1}. \end{aligned} \end{aligned}$$
(A.6)

It follows from (A.6) that

$$\begin{aligned} \begin{aligned} E_q(\underline{X_{1,2}X_{2,5}})&= q F_q(\underline{X_{1,2}})*F_q(\underline{X_{2,5}}) = L_q(\underline{X_{1,2}X_{2,5}}) + q^2 L_q(\underline{X_{2,3}}), \end{aligned} \end{aligned}$$
(A.7)

where \(L_q(\underline{X_{1,2}X_{2,5}}) = F_q(\underline{X_{1,2}X_{2,5}})+F_q(\underline{X_{2,3}})\) and \(P_{X_{1,2}X_{2,5},\,X_{2,3}}(q) = q^2 \in q\mathbb {Z}[q]\). As in Example A.1, it follows from (A.6) and (A.7) that the quantum positivity of \(L_q(\underline{X_{1,2}X_{2,5}})\) holds. Note that \(L_q(\underline{X_{1,2}X_{2,5}})\) has two dominant monomials.

Let us also consider \(E_q(\underline{X_{1,2}X_{2,5}^2})\). By (A.6), \(E_q(\underline{X_{1,2}X_{2,5}^2})\) has three dominant monomials, namely,

$$\begin{aligned} \underline{X_{1,2}X_{2,5}^2} = q^4 \widetilde{X}_{1,2}*\widetilde{X}_{2,5}^2, \quad (q^2 + q^4) \widetilde{X}_{2,3} * \widetilde{X}_{2,5} = (q+q^3) \underline{X_{2,3}X_{2,5}}, \quad q^5 \widetilde{X}_{1,4} = q^4 \underline{X_{1,4}}. \end{aligned}$$

Then we have

$$\begin{aligned} \begin{aligned}&E_q(\underline{X_{1,2}X_{2,5}^2}) = L_q(\underline{X_{1,2}X_{2,5}^2}) + (q+q^3) L_q(\underline{X_{2,3}X_{2,5}}) + q^4 L_q(\underline{X_{1,4}}), \end{aligned} \end{aligned}$$
(A.8)

where \(L_q(\underline{X_{1,2}X_{2,5}^2}) = F_q(\underline{X_{1,2}X_{2,5}^2})\), \(L_q(\underline{X_{2,3}X_{2,5}}) = F_q(\underline{X_{2,3}X_{2,5}})\), and

$$\begin{aligned} P_{X_{1,2}X_{2,5}^2,\, X_{2,3}X_{2,5}}(q) = q+q^3, \quad P_{X_{1,2}X_{2,5}^2,\, X_{1,4}}(q) = q^4 \in q\mathbb {Z}[q]. \end{aligned}$$

We provide the formula of \(L_q(\underline{X_{2,3}X_{2,5}}) = F_q(\underline{X_{2,3}X_{2,5}})\) as follows:

(A.9)

Then one may compute the formula of \(L_q(\underline{X_{1,2}X_{2,5}^2})\) by using (A.8) with (A.6) and (A.9) (or the q-algorithm directly), and then the quantum positivity of \(L_q(\underline{X_{1,2}X_{2,5}^2})\) also follows.

1.3 Positivity of Kazhdan–Lusztig polynomials

This subsection presents examples for the positivity of the KL-type polynomials \(P_{m,m'}(q) \in q\mathbb {Z}[q]\) with at least 2 terms.

Example A.3

In Example A.2 (for type \(B_2\)), we have seen the positivity of KL-type polynomial given by

$$\begin{aligned} P_{X_{1,2}X_{2,5}^2,\, X_{2,3}X_{2,5}}(q) = q+q^3 \in q\mathbb {Z}[q], \end{aligned}$$

which is an example for the positivity of KL-type polynomials with 2-terms. Let us consider the case of type \(G_2\) to investigate more complicated examples.

For \(\underline{m}= \underline{X_{2,0}X_{1,5}^2}\), we have

$$\begin{aligned} \begin{aligned} E_q(\underline{m})&= L_q(\underline{m}) + (q^2 + q^4) L_q(\underline{X_{1,1}X_{1,5}}) + q^6 L_q(\underline{X_{1,3}}), \end{aligned} \end{aligned}$$

where \(L_q(\underline{m}) = F_q(\underline{m}) + L_q(\underline{X_{1,1}X_{1,5}})\) and \(L_q(\underline{X_{1,1}X_{1,5}}) = F_q(\underline{X_{1,1}X_{1,5}}) + L_q(\underline{X_{1,3}})\) and the KL-type polynomials are

$$\begin{aligned} P_{X_{2,0}X_{1,5}^2,\, X_{1,1}X_{1,5}}(q) = q^2 + q^4, \quad P_{X_{2,0}X_{1,5}^2,\, X_{1,3}}(q) = q^6 \in q\mathbb {Z}[q]. \end{aligned}$$

For \(\underline{m}= \underline{X_{2,0}^2 X_{1,1} X_{1,3}}\), we have

$$\begin{aligned} \begin{aligned} E_q(\underline{m})&= L_q(\underline{m}) + q L_q(\underline{X_{2,0}^2X_{2,2}}) + \left( 2q^4 + q^6 + q^8 + q^{10} \right) L_q(\underline{X_{2,0}X_{1,1}^3}), \end{aligned} \end{aligned}$$

where \(L_q(\underline{m}) = F_q(\underline{m}) + (q^{-2} + 1 + q^2) F_q(\underline{X_{2,0}X_{1,1}^3})\) and the KL-type polynomials are

$$\begin{aligned} P_{X_{2,0}^2 X_{1,1} X_{1,3},\, X_{2,0}^2 X_{2,2}}(q) = q, \quad P_{X_{2,0}^2 X_{1,1} X_{1,3},\, X_{2,0}X_{1,1}^3}(q) = 2q^4 + q^6 + q^8 + q^{10} \in q\mathbb {Z}[q]. \end{aligned}$$

For \(\underline{m}= \underline{X_{2,0}^2 X_{2,4}}\), we have the expansion of \(E_q(\underline{m})-L_q(\underline{m})\) in terms of \(\textsf{L}_q\) as follows:

where \(L_q(\underline{X_{2,0}X_{1,1}X_{1,3}}) = F_q(\underline{X_{2,0}X_{2,2}}) + (q^{-2} + 1 + q^2) L_q(\underline{X_{1,1}^3})\) and the KL-type polynomials (in \(q\mathbb {Z}[q]\)) are

$$\begin{aligned} \begin{aligned}&P_{X_{2,0}^2 X_{2,4},\, X_{2,0}X_{1,1}X_{1,3}}(q) = q^4 + q^6 + q^8, \quad P_{X_{2,0}^2 X_{2,4},\, X_{2,0}X_{2,2}}(q) = q^3 + q^6,\\&P_{X_{2,0}^2 X_{2,4},\, X_{1,1}^3}(q) = 2q^2 + 6q^4 + 6q^6 +4q^8 + 2q^{10} + q^{12}. \end{aligned} \end{aligned}$$

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Jang, IS., Lee, KH. & Oh, Sj. Quantization of Virtual Grothendieck Rings and Their Structure Including Quantum Cluster Algebras. Commun. Math. Phys. 405, 173 (2024). https://doi.org/10.1007/s00220-024-05037-z

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