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Auslander–Reiten quiver and representation theories related to KLR-type Schur–Weyl duality

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Abstract

We introduce new partial orders on the sequence positive roots and study the statistics of the poset by using Auslander–Reiten quivers for finite type ADE. Then we can prove that the statistics provide interesting information on the representation theories of KLR-algebras, quantum groups and quantum affine algebras including Dorey’s rule, bases theory for quantum groups, and denominator formulas between fundamental representations. As applications, we prove Dorey’s rule for quantum affine algebras \(U_q(E_{6,7,8}^{(1)})\) and partial information of denominator formulas for \(U_q(E_{6,7,8}^{(1)})\). We also suggest conjecture on complete denominator formulas for \(U_q(E_{6,7,8}^{(1)})\).

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Notes

  1. Recall that a cover of x in a poset P with partial order \(\prec \) is an element \(y \in P\) such that \(x \prec y\) and there does not exists \(y' \in P\) such that \(x \prec y' \prec y\).

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Acknowledgements

The author would like to express his sincere gratitude to Professor Masaki Kashiwara, Myungho Kim and Chul-hee Lee for many fruitful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Ewha Womans University Seoul, Seoul, 120-750, Korea

    Se-jin Oh

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  1. Se-jin Oh
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Correspondence to Se-jin Oh.

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This work was supported by NRF Grant #2016R1C1B2013135.

Appendices

Appendix A: Dynkin quiver Q of type \(E_6\), its AR-quiver \(\Gamma _Q\) and Dorey’s rule for \(E_6^{(1)}\)

(1) Let us consider the Dynkin quiver Q given as follows:

Its AR-quiver \(\Gamma _Q\) can be drawn as follows :

By reading pairs and their socles, we can obtain Dorey’s rule for \(E_6^{(1)}\). Here we list several Dorey’s type morphisms:

$$\begin{aligned}&V(\varpi _6) \hookrightarrow V(\varpi _6)_{ q^{4}} \otimes V(\varpi _6)_{q^{-4}}, \quad V(\varpi _3) \hookrightarrow V(\varpi _1)_{q^{2}} \otimes V(\varpi _2)_{-q^{-1}}, \\&\quad V(\varpi _4) \hookrightarrow V(\varpi _1)_{-q^{3}} \otimes V(\varpi _6)_{q^{-2}},\\&V(\varpi _6) \hookrightarrow V(\varpi _1)_{-q^{3}} \otimes V(\varpi _5)_{-q^{-3}}, \quad V(\varpi _5) \hookrightarrow V(\varpi _2)_{-q^{9}} \otimes V(\varpi _3)_{q^{-2}}, \\&\quad V(\varpi _5) \hookrightarrow V(\varpi _1)_{q^{4}} \otimes V(\varpi _1)_{q^{-4}},\\&V(\varpi _4) \hookrightarrow V(\varpi _1)_{-q^{9}} \otimes V(\varpi _3)_{-q^{-1}}, \quad V(\varpi _2) \hookrightarrow V(\varpi _1)_{-q^{5}} \otimes V(\varpi _4)_{q^{-2}},\\&\quad V(\varpi _1) \hookrightarrow V(\varpi _1)_{-q^{5}} \otimes V(\varpi _6)_{-q^{-5}},\\&V(\varpi _5) \hookrightarrow V(\varpi _2)_{-q^{5}} \otimes V(\varpi _6)_{-q^{-5}}, \quad V(\varpi _3) \hookrightarrow V(\varpi _3)_{q^{4}} \otimes V(\varpi _3)_{q^{-4}}, \\&\quad V(\varpi _6) \hookrightarrow V(\varpi _3)_{-q^{5}} \otimes V(\varpi _3)_{-q^{-5}},\\&V(\varpi _6)_{q^{2}} \otimes V(\varpi _5)_{-q^{1}} \hookrightarrow V(\varpi _1)_{-q^{5}} \otimes V(\varpi _4), \quad V(\varpi _6)_{q^{2}} \otimes V(\varpi _4)_{q^{2}} \hookrightarrow V(\varpi _2)_{-q^{4}} \otimes V(\varpi _2),\\&V(\varpi _1)_{q^{6}} \otimes V(\varpi _6)_{q^{4}} \hookrightarrow V(\varpi _2)_{-q^{7}} \otimes V(\varpi _1), \quad V(\varpi _6)_{-q^{3}} \otimes V(\varpi _4)_{-q^{3}} \hookrightarrow V(\varpi _3)_{q^{4}} \otimes V(\varpi _1),\\&V(\varpi _1)_{-q^{1}} \otimes V(\varpi _1)_{-q^{5}} \otimes V(\varpi _5)_{-q^{3}} \hookrightarrow V(\varpi _2)_{q^{6}} \otimes V(\varpi _2). \end{aligned}$$

(2) The convex order \(\prec _{[{\widetilde{w}_0}]}\) in (b) of Example 1.10 can be visualized by the result of [32] as follows:

Appendix B: Dynkin quiver Q of type \(E_7\), its AR-quiver \(\Gamma _Q\) and Dorey’s rule for \(E_7^{(1)}\)

Let us consider the Dynkin quiver Q given as follows:

Its AR-quiver \(\Gamma _Q\) can be drawn as follows:

By reading pairs and their socles, we can obtain Dorey’s rule for \(E_7^{(1)}\). Here we list several Dorey’s type morphisms:

$$\begin{aligned} V(\varpi _4)&\hookrightarrow V(\varpi _3)_{-q^{1}} \otimes V(\varpi _1)_{q^{-2}}, \quad V(\varpi _1) \hookrightarrow V(\varpi _3)_{-q^{3}} \otimes V(\varpi _1)_{q^{-10}}, \\ V(\varpi _3)&\hookrightarrow V(\varpi _4)_{-q^{1}} \otimes V(\varpi _1)_{-q^{-15}},\\ V(\varpi _2)&\hookrightarrow V(\varpi _5)_{q^{2}} \otimes V(\varpi _1)_{-q^{-13}}, \quad V(\varpi _1) \hookrightarrow V(\varpi _6)_{q^{4}} \otimes V(\varpi _1)_{q^{-10}}, \\ V(\varpi _7)&\hookrightarrow V(\varpi _7)_{q^{8}} \otimes V(\varpi _1)_{-q^{-5}},\\ V(\varpi _6)&\hookrightarrow V(\varpi _1)_{q^{4}} \otimes V(\varpi _3)_{q^{-4}}, \quad V(\varpi _5) \hookrightarrow V(\varpi _2)_{q^{2}} \otimes V(\varpi _1)_{-q^{-3}},\\ V(\varpi _3)&\hookrightarrow V(\varpi _3)_{q^{6}} \otimes V(\varpi _3)_{q^{-6}},\\ V(\varpi _6)&\hookrightarrow V(\varpi _4)_{q^{4}} \otimes V(\varpi _3)_{-q^{-11}}, \quad V(\varpi _1) \hookrightarrow V(\varpi _4)_{q^{8}} \otimes V(\varpi _4)_{q^{-8}}, \\ V(\varpi _4)&\hookrightarrow V(\varpi _4)_{q^{6}} \otimes V(\varpi _4)_{q^{-6}},\\ V(\varpi _2)_{-q^{7}}&\otimes V(\varpi _7)_{-q^{5}} \hookrightarrow V(\varpi _3)_{-q^{9}} \otimes V(\varpi _1), \quad V(\varpi _1)_{q^{6}} \otimes V(\varpi _7)_{-q^{5}} \hookrightarrow V(\varpi _2)_{-q^{9}} \otimes V(\varpi _1),\\ V(\varpi _1)_{-q^{3}}&\otimes V(\varpi _7)_{q^{4}} \hookrightarrow V(\varpi _2)_{q^{6}} \otimes V(\varpi _1), \quad V(\varpi _3)_{q^{2}} \otimes V(\varpi _2)_{q^{2}} \hookrightarrow V(\varpi _6)_{-q^{5}} \otimes V(\varpi _5),\\ V(\varpi _7)_{-q^{1}}&\otimes V(\varpi _7)_{-q^{7}} \otimes V(\varpi _1)_{q^{4}} \hookrightarrow V(\varpi _6)_{q^{8}} \otimes V(\varpi _6). \end{aligned}$$

Appendix C: Dynkin quiver Q of type \(E_8\) and its AR-quiver \(\Gamma _Q\)

Let us consider the Dynkin quiver Q given as follows:

Note that \(i^*=i\) for all \(i\in I\). Its AR-quiver \(\Gamma _Q\) can be drawn as follows:

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Oh, Sj. Auslander–Reiten quiver and representation theories related to KLR-type Schur–Weyl duality. Math. Z. 291, 499–554 (2019). https://doi.org/10.1007/s00209-018-2093-2

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