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The (q, t)-Cartan matrix specialized at \(q=1\) and its applications

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Abstract

The (q, t)-Cartan matrix specialized at \(t=1\), usually called the quantum Cartan matrix, has deep connections with (i) the representation theory of its untwisted quantum affine algebra, and (ii) quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetric type. In this paper, we study the (q, t)-Cartan matrix specialized at \(q=1\), called the t-quantized Cartan matrix, and investigate the relations with (ii\('\)) its corresponding unipotent quantum coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type.

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Notes

  1. In [15],  (3.10) was called \(\mathfrak {g}\)-additive property.

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Acknowledgements

The second author is grateful to I.-S. Jang, Y.-H. Kim, K.-H. Lee and R. Fujita for helpful discussions.

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Authors and Affiliations

  1. Kyoto University Institute for Advanced Study, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Masaki Kashiwara

  2. Korea Institute for Advanced Study, Seoul, 02455, Korea

    Masaki Kashiwara

  3. Ewha Womans University Seoul, 52 Ewhayeodae-gil, Daehyeon-dong, Seodaemun-gu, Seoul, South Korea

    Se-jin Oh

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The research of M. Kashiwara was supported by Grant-in-Aid for Scientific Research (B) 20H01795, Japan Society for the Promotion of Science.

The research of S.-j. Oh was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2022R1A2C1004045).

Appendix A: \(\widetilde{\mathfrak {d}}_{i,j}(t)\) for \(E_7\) and \(E_8\)

Appendix A: \(\widetilde{\mathfrak {d}}_{i,j}(t)\) for \(E_7\) and \(E_8\)

1.1 A.1 \(E_7\)

Here is the list of \(\widetilde{\mathfrak {d}}_{i,j}(t)\) for \(E_7\).

$$\begin{aligned}&\widetilde{\mathfrak {d}}_{1,1}(t) = t^1+t^{7}+t^{11}+t^{17}, \qquad \widetilde{\mathfrak {d}}_{1,2}(t) = t^4+t^{8}+t^{10}+t^{14}, \\&\widetilde{\mathfrak {d}}_{1,3}(t)=t^2+t^{6}+t^{8}+t^{10}+t^{12}+t^{16}, \ \qquad \widetilde{\mathfrak {d}}_{1,4}(t)=t^3+t^{5}+t^{7}+2t^{9}+t^{11}+t^{13}+t^{15},\\&\widetilde{\mathfrak {d}}_{1,6}(t)=t^5+t^{7}+t^{11}+t^{13}, \qquad \widetilde{\mathfrak {d}}_{1,7}(t)=t^6+t^{12}, \\&\widetilde{\mathfrak {d}}_{2,2}(t)=t^1+t^{5}+t^{7}+t^{9}+t^{11}+t^{13}+t^{17}, \qquad \widetilde{\mathfrak {d}}_{2,3}(t)=\widetilde{\mathfrak {d}}_{1,4}(t) \\&\widetilde{\mathfrak {d}}_{2,4}(t)=t^2+t^{4}+2t^{6}+2t^{8}+2t^{10}+2t^{12}+t^{14}+t^{16}, \\&\widetilde{\mathfrak {d}}_{2,5}(t)=t^3+t^{5}+2t^{7}+t^{9}+2t^{11}+t^{13}+t^{15}, \ \widetilde{\mathfrak {d}}_{2,7}(t)=t^{5}+t^{9}+t^{13}, \\&\widetilde{\mathfrak {d}}_{2,6}(t)=t^{4}+t^{6}+t^{8}+t^{10}+t^{12}+t^{14}, \\&\widetilde{\mathfrak {d}}_{3,3}(t) = t^1+t^{3}+t^{5}+2t^{7}+2t^{9}+2t^{11}+t^{13}+t^{15}+t^{17}, \\&\widetilde{\mathfrak {d}}_{3,4}(t) = t^2+2t^{4}+2t^{6}+3t^{8}+3t^{10}+2t^{12}+2t^{14}+t^{16}, \\&\widetilde{\mathfrak {d}}_{3,5}(t) = t^{3}+2t^{5}+2t^{7}+2t^{9}+2t^{11}+2t^{13}+t^{15}, \\&\widetilde{\mathfrak {d}}_{3,6}(t) = t^{4}+2t^{6}+t^{8}+t^{10}+2t^{12}+t^{14}, \qquad \widetilde{\mathfrak {d}}_{3,7}(t) = t^{5}+t^{7}+t^{11}+t^{13}, \\&\widetilde{\mathfrak {d}}_{4,4}(t) = t^1+2t^{3}+3t^{5}+4t^{7}+4t^{9}+4t^{11}+3t^{13}+2t^{15}+t^{17}, \\&\widetilde{\mathfrak {d}}_{4,5}(t) = t^2+2t^{4}+3t^{6}+3t^{8}+3t^{10}+3t^{12}+2t^{14}+t^{16}, \\&\widetilde{\mathfrak {d}}_{4,6}(t) =\widetilde{\mathfrak {d}}_{3,5}(t), \qquad \widetilde{\mathfrak {d}}_{4,7}(t) = t^{4}+t^{6}+t^{8}+t^{10}+ t^{12}+ t^{14}, \\&\widetilde{\mathfrak {d}}_{5,5}(t) = t^1+t^{3}+2t^{5}+2t^{7}+3t^{9}+2t^{11}+2t^{13}+t^{15}+t^{17}, \\&\widetilde{\mathfrak {d}}_{5,6}(t) = t^2+t^{4}+t^{6}+2t^{8}+2t^{10}+t^{12}+t^{14}+t^{16}, \quad \widetilde{\mathfrak {d}}_{5,7}(t) = t^{3}+t^{7}+t^{11}+t^{15}, \\&\widetilde{\mathfrak {d}}_{6,6}(t) = t^1+t^{3}+t^{7}+2t^{9}+t^{11}+t^{15}+t^{17}, \qquad \widetilde{\mathfrak {d}}_{6,7}(t) = t^2+t^{8}+t^{10}+t^{16}, \\&\widetilde{\mathfrak {d}}_{7,7}(t) = t^1+t^{9}+t^{17} \qquad \text { and } \qquad \widetilde{\mathfrak {d}}_{i,j}(t) = \widetilde{\mathfrak {d}}_{j,i}(t). \end{aligned}$$

1.2 A.2 \(E_8\)

Here is the list of \(\widetilde{\mathfrak {d}}_{i,j}(t)\) for \(E_8\).

$$\begin{aligned} \widetilde{\mathfrak {d}}_{1,1}(t)&= t^1+t^{7}+t^{11}+t^{13}+t^{17}+t^{19}+t^{23}+t^{29}, \\ \widetilde{\mathfrak {d}}_{1,2}(t)&= t^4+t^{8}+t^{10}+t^{12}+t^{14}+t^{16}+t^{18}+t^{20}+t^{22}+t^{26}, \\ \widetilde{\mathfrak {d}}_{1,3}(t)&= t^2+t^{6}+t^{8}+t^{10}+2t^{12}+t^{14}+t^{16}+2t^{18}+t^{20}+t^{22}+t^{24}+t^{28}, \\ \widetilde{\mathfrak {d}}_{1,4}(t)&= t^3+t^{5}+t^{7}+2t^{9}+2t^{11}+2t^{13}+2t^{15}+2t^{17}+2t^{19}+2t^{21}+t^{23}+t^{25}+t^{27}, \\ \widetilde{\mathfrak {d}}_{1,5}(t)&= t^4+t^{6}+t^{8}+2t^{10}+t^{12}+2t^{14}+2t^{16}+t^{18}+2t^{20}+t^{22}+t^{24}+t^{26}, \\ \widetilde{\mathfrak {d}}_{1,6}(t)&= t^{5}+t^{7}+t^{9}+t^{11}+t^{13}+2t^{15}+t^{17}+t^{19}+t^{21}+t^{23}+t^{25}, \\ \widetilde{\mathfrak {d}}_{1,7}(t)&= t^{6}+t^{8}+t^{12}+t^{14}+t^{16}+t^{18}+t^{22}+t^{24}, \\ \widetilde{\mathfrak {d}}_{1,8}(t)&= t^{7}+t^{13}+t^{17}+t^{23}, \\ \widetilde{\mathfrak {d}}_{2,2}(t)&= t^1+t^{5}+t^{7}+t^{9}+2t^{11}+t^{13}+2t^{15}+t^{17}+2t^{19}+t^{21}+t^{23}+t^{25}+t^{29}, \\ \widetilde{\mathfrak {d}}_{2,3}(t)&= t^3+t^{5}+t^{7}+2(t^{9}+t^{11}+t^{13}+t^{15}+t^{17}+t^{19}+t^{21})+t^{23}+t^{25}+t^{27}, \\ \widetilde{\mathfrak {d}}_{2,4}(t)&= t^2+t^{4}+2t^{6}+2t^{8}+3(t^{10}+t^{12}+t^{14}+t^{16}+t^{18}+t^{20})+2t^{22}+2t^{24}+t^{26}+t^{28}, \\ \widetilde{\mathfrak {d}}_{2,5}(t)&= t^3+t^{5}+2t^{7}+2t^{9}+2t^{11}+3t^{13}+3t^{15}+3t^{17}+2t^{19}+2t^{21}+2t^{23}+t^{25}+t^{27}, \\ \widetilde{\mathfrak {d}}_{2,6}(t)&= t^4+t^{6}+2t^{8}+t^{10}+2t^{12}+2t^{14}+2t^{16}+2t^{18}+t^{20}+2t^{22}+t^{24}+t^{26}, \\ \widetilde{\mathfrak {d}}_{2,7}(t)&=\widetilde{\mathfrak {d}}_{1,6}(t), \qquad \qquad \qquad \widetilde{\mathfrak {d}}_{2,8}(t) = t^{6}+t^{10}+t^{14}+t^{16}+t^{20}+t^{24}, \\ \widetilde{\mathfrak {d}}_{3,3}(t)&= t^1+t^{3}+t^{5}+2t^{7}+2t^{9}+3t^{11}+3t^{13}+2t^{15}+3t^{17}+3t^{19}+2t^{21}+2t^{23} \\&\quad +t^{25}+t^{27}+t^{29}, \\ \widetilde{\mathfrak {d}}_{3,4}(t)&= t^2+2(t^{4}+t^{6})+3t^{8}+4(t^{10}+t^{12}+t^{14}+t^{16}+t^{18}+t^{20})+3t^{22}+2(t^{24}+t^{26})+t^{28}, \\ \widetilde{\mathfrak {d}}_{3,5}(t)&= t^3+2t^{5}+2t^{7}+3t^{9}+3t^{11}+3t^{13}+4t^{15}+3t^{17}+3t^{19}+3t^{21}+2t^{23}+2t^{25}+t^{27}, \\ \widetilde{\mathfrak {d}}_{3,6}(t)&= t^4+2t^{6}+2t^{8}+2t^{10}+2t^{12}+3t^{14}+3t^{16}+2t^{18}+2t^{20}+2t^{22}+2t^{24}+t^{26}, \\ \widetilde{\mathfrak {d}}_{3,7}(t)&= t^{5}+2t^{7}+t^{9}+t^{11}+2t^{13}+2t^{15}+2t^{17}+t^{19}+t^{21}+2t^{23}+t^{25}, \\ \widetilde{\mathfrak {d}}_{3,8}(t)&= t^{6}+t^{8}+t^{12}+t^{14}+t^{16}+t^{18}+t^{22}+t^{24}, \\ \widetilde{\mathfrak {d}}_{4,4}(t)&= t^1+2t^{3}+3t^{5}+4t^{7}+5t^{9}+6t^{11}+6t^{13}+6t^{15}+6t^{17}+6t^{19}+5t^{21}\\&\quad +4t^{23} +3t^{25}+2t^{27}+t^{29}, \\ \widetilde{\mathfrak {d}}_{4,5}(t)&= t^2+2t^{4}+3t^{6}+4t^{8}+4t^{10}+5t^{12}+5t^{14}+5t^{16}+5t^{18}+4t^{20}+4t^{22}\\&\quad +3t^{24}+2t^{26}+t^{28}, \\ \widetilde{\mathfrak {d}}_{4,6}(t)&= t^3+2t^{5}+3t^{7}+3t^{9}+3t^{11}+4t^{13}+4t^{15}+4t^{17}+3t^{19}+3t^{21}+3t^{23}+2t^{25}+t^{27}, \\ \widetilde{\mathfrak {d}}_{4,7}(t)&= \widetilde{\mathfrak {d}}_{3,6}(t), \ \ \widetilde{\mathfrak {d}}_{4,8}(t) = t^{5}+t^{7}+t^{9}+t^{11}+t^{13}+2t^{15}+t^{17}+t^{19}+t^{21}+t^{23}+t^{25}, \\ \widetilde{\mathfrak {d}}_{5,5}(t)&= t^1+t^{3}+2t^{5}+3t^{7}+3t^{9}+4t^{11}+4t^{13}+4t^{15}+4t^{17}+4t^{19}+3t^{21}\\&\quad +3t^{23}+2t^{25}+t^{27}+t^{29}, \\ \widetilde{\mathfrak {d}}_{5,6}(t)&= t^2+t^{4}+2t^{6}+2t^{8}+3t^{10}+3t^{12}+3t^{14}+3t^{16}+3t^{18}+3t^{20}+2t^{22}\\&\quad +2t^{24}+t^{26}+t^{28}, \\ \widetilde{\mathfrak {d}}_{5,7}(t)&= t^3+t^{5}+t^{7}+2t^{9}+2t^{11}+2t^{13}+2t^{15}+2t^{17}+2t^{19}+2t^{21}+t^{23}+t^{25}+t^{27}, \\ \widetilde{\mathfrak {d}}_{5,8}(t)&= t^{4}+t^{8}+t^{10}+t^{12}+t^{14}+t^{16}+t^{18}+t^{20}+t^{22}+t^{26}, \\ \widetilde{\mathfrak {d}}_{6,6}(t)&= t^1+t^{3}+t^{5}+t^{7}+2t^{9}+3t^{11}+2t^{13}+2t^{15}+2t^{17}+3t^{19}+2t^{21}\\&\quad +t^{23}+t^{25}+t^{27}+t^{29}, \\ \widetilde{\mathfrak {d}}_{6,7}(t)&= t^2+t^{4}+t^{8}+2t^{10}+2t^{12}+t^{14}+t^{16}+2t^{18}+2t^{20}+t^{22}+t^{26}+t^{28}, \\ \widetilde{\mathfrak {d}}_{6,8}(t)&= t^3+t^{9}+t^{11}+t^{13}+t^{17}+t^{19}+t^{21}+t^{27}, \\ \widetilde{\mathfrak {d}}_{7,7}(t)&= t^1+t^{3}+t^{9}+2t^{11}+t^{13}+t^{17}+2t^{19}+t^{21}+t^{27}+t^{29}, \\ \widetilde{\mathfrak {d}}_{7,8}(t)&= t^2+t^{10}+t^{12}+t^{18}+t^{20}+t^{28}, \\ \widetilde{\mathfrak {d}}_{8,8}(t)&= t^1+t^{11}+t^{19}+t^{29} \qquad \qquad \quad \text {and} \qquad \qquad \quad \widetilde{\mathfrak {d}}_{i,j}(t) = \widetilde{\mathfrak {d}}_{j,i}(t). \end{aligned}$$

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Kashiwara, M., Oh, Sj. The (q, t)-Cartan matrix specialized at \(q=1\) and its applications. Math. Z. 303, 42 (2023). https://doi.org/10.1007/s00209-022-03195-1

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