Correction to: Categorical Relations Between Langlands Dual Quantum Affine Algebras: Exceptional Cases

1 Correction to: Commun. Math. Phys. 368, 295–367 (2019)                  https://doi.org/10.1007/s00220-019-03287-w

2 Introduction

In this short note, we remove the ambiguities in our paper [2, Prop. 4.29, Prop. 7.8, App. C, App. D]. We will use the same notation as in [2]; in particular, $d_{i,j}(z) \mathbin {:=}d_{i,j}^{X_n^{(r)}}(z)$ denotes the denominator formula for $R^{\mathrm {norm}}_{V(\varpi _i)_x,V(\varpi _j)_y}$ in the affine Lie algebra of type $X_n^{(r)}$, where $z = x/y$. We also require the following.

Proposition 1.1

[3, Prop. 6.8]. Let Q be a Dynkin quiver of finite simply laced type. For a [Q]-minimal pair $(\alpha ,\beta )$ of a simple sequence $\underline{s}=(\alpha _1,\ldots ,\alpha _k)$, $V^{(1)}_Q(\alpha _1) \otimes \cdots \otimes V^{(1)}_Q(\alpha _k)$ is simple, and there exists a surjective homomorphism

$$\begin{aligned} V_Q^{(1)}(\beta ) \otimes V_Q^{(1)}(\alpha ) \twoheadrightarrow V^{(1)}_Q(\alpha _1) \otimes \cdots \otimes V^{(1)}_Q(\alpha _k). \end{aligned}$$

3 Type $E_6^{(1)}$ and $E_6^{(2)}$

In [2, Prop. 4.29], there was the ambiguity

$$\begin{aligned} d_{4,4}^{E_6^{(1)}}(z) = (z-q^2)(z-q^4)^2(z-q^6)^{2+\epsilon }(z-q^8)^3(z-q^{10})^2(z-q^{12}) \end{aligned}$$

(2.1)

for some $\epsilon \in \{0,1\}$. We shall show that $\epsilon = 1$.

Proposition 2.1

For type $E_6^{(1)}$, we have

$$\begin{aligned} d_{4,4}(z) = (z-q^2)(z-q^4)^2(z-q^6)^3(z-q^8)^3(z-q^{10})^2(z-q^{12}). \end{aligned}$$

Proof

Note that, from the AR-quiver of type $E_6$ in [2, (2.6)], we can read that

Hence Proposition 1.1 states that we have a homomorphism

$$\begin{aligned} V(\varpi _1)_{(-q)^{-3}} \otimes V(\varpi _4)_{(-q)^1} \twoheadrightarrow V(\varpi _2)\otimes V(\varpi _5). \end{aligned}$$

(2.2)

Then we have

$$\begin{aligned} \dfrac{ d_{1,4}((-q)^{-3}z)d_{4,4}((-q)^{1}z)}{d_{2,4}(z)d_{5,4}(z)} \times \dfrac{ a_{2,4}(z)d_{5,4}(z)}{ a_{1,4}((-q)^{-3}z)a_{4,4}((-q)^{1}z)}\in \mathbf {k}[z^{\pm 1}] \end{aligned}$$

(2.3)

by [1, Lemma C.15]. From [2, Lemma 3.9], one can compute that

$$\begin{aligned}&a_{2,4}(z) \equiv \dfrac{[19][5][23][1]}{[7][17][11][13]}&a_{3,4}(z) \equiv a_{4,5}(z) \equiv \dfrac{[1][3][21][23]}{[9][11][13][15]} \\&a_{1,4}(z) \equiv \dfrac{[2][22]}{[10][14]}&a_{4,4}(z) \equiv \dfrac{[0][2][4][20][22][24]}{[8][10][12]^2[14][16]}. \end{aligned}$$

Since we have computed $d_{1,4}(z)$, $d_{2,4}(z)$ and $d_{5,4}(z)$,  (2.3) can be written as follows:

$$\begin{aligned}&\dfrac{ d_{1,4}((-q)^{-3}z)d_{4,4}((-q)^{1}z)}{d_{2,4}(z)d_{5,4}(z)} = \dfrac{ (z+q^1)(z+q^3)^2(z+q^5)^{2\text { or } 3}(z+q^7)^4(z-q^{9})^3(z-q^{11})^2 (z+q^{13}) }{ (z+q^3)^2(z+q^5)^3(z+q^7)^4(z+q^9)^3(z+q^{11})^2} \end{aligned}$$

by (2.1) and

$$\begin{aligned} \dfrac{ a_{2,4}(z)d_{5,4}(z)}{ a_{1,4}((-q)^{-3}z)a_{4,4}((-q)^{1}z)}&= \dfrac{[19][5][23][1]}{[7][17][11][13]} \times \dfrac{[1][3][21][23]}{[9][11][13][15]}. \times \dfrac{[7][11]}{[-1][19]} \times \dfrac{[9][11][13]^2[15][17]}{[1][3][5][21][23][25]} \\&= \dfrac{[23][1]}{[-1][25]} = \dfrac{(z-(-q)^{-1})}{(z-(-q)^{1})}. \end{aligned}$$

Thus we have

$$\begin{aligned}&\dfrac{ (z+q^1)(z+q^3)^2(z+q^5)^{2\text { or } 3}(z+q^7)^4(z-q^{9})^3(z-q^{11})^2 (z+q^{13}) }{ (z+q^3)^2(z+q^5)^3(z+q^7)^4(z+q^9)^3(z+q^{11})^2} \times \dfrac{(z-(-q)^{-1})}{(z-(-q)^{1})} \\&= \dfrac{ (z+q^{-1})(z+q^3)^2(z+q^5)^{2\text { or } 3}(z+q^7)^4(z-q^{9})^3(z-q^{11})^2 (z+q^{13}) }{ (z+q^3)^2(z+q^5)^3(z+q^7)^4(z+q^9)^3(z+q^{11})^2} \in \mathbf {k}[z^{\pm 1}] \end{aligned}$$

which implies that the order of degree $(-q)^6$ should be 3. $\square$

By applying generalized Schur–Weyl duality

we have the twisted analogue of Proposition 1.1:

Proposition 2.2

Let Q be a Dynkin quiver of finite simply-laced type. For a [Q]-minimal pair $(\alpha ,\beta )$ of a simple sequence $\underline{s}=(\alpha _1,\ldots ,\alpha _k)$, $V^{(2)}_Q(\alpha _1) \otimes \cdots \otimes V^{(2)}_Q(\alpha _k)$ is simple, and there exists a surjective homomorphism

$$\begin{aligned} V_Q^{(2)}(\beta ) \otimes V_Q^{(2)}(\alpha ) \twoheadrightarrow V^{(2)}_Q(\alpha _1) \otimes \cdots \otimes V^{(2)}_Q(\alpha _k). \end{aligned}$$

In particular, (2.2) transfers to the following homomorphism in $\mathcal {C}^{(2)}_Q$ under the map in [2, Prop. 6.5]:

$$\begin{aligned} V(\varpi _1)_{(-q)^{-3}} \otimes V(\varpi _3)_{\sqrt{-1}(-q)^1} \twoheadrightarrow V(\varpi _2)_{-1} \otimes V(\varpi _4)_{\sqrt{-1}}. \end{aligned}$$

(2.4)

Next, we resolve from [2, Prop. 7.8] the ambiguity in type $E_6^{(2)}$

$$\begin{aligned} d_{3,3}^{E_6^{(2)}}(z)= (z^2-q^4)(z^2-q^8)^{2}(z^2-q^{12})^{2+\epsilon '}(z^2-q^{16})^{3}(z^2-q^{20})^2(z^2-q^{24}) \nonumber \\ \end{aligned}$$

(2.5)

for some $\epsilon ' \in \{0,1\}$ by showing $\epsilon ' = 1$.

Proposition 2.3

For type $E_6^{(2)}$, we have

$$\begin{aligned} d_{3,3}(z)= (z^2-q^4)(z^2-q^8)^{2}(z^2-q^{12})^{3}(z^2-q^{16})^{3}(z^2-q^{20})^2(z^2-q^{24}). \end{aligned}$$

Proof

By (2.4) and [1, Lemma C.15], we have

$$\begin{aligned} \dfrac{ d_{1,3}(\sqrt{-1}(-q)^{-3}z)d_{3,3}((-q)^{1}z)}{d_{2,3}(\sqrt{-1}z)d_{3,4}(z)} \times \dfrac{ a_{2,3}(\sqrt{-1}z)a_{3,4}(z)}{ a_{1,3}(\sqrt{-1}(-q)^{-3}z)a_{3,3}((-q)^{1}z)}\in \mathbf {k}[z^{\pm 1}]. \end{aligned}$$

(2.6)

From [2, Lemma 3.9], one can compute that

$$\begin{aligned}&a_{2,3}(\sqrt{-1}z) \equiv \dfrac{ \{ 3 \} \{ 21 \} \{ 1 \} \{ 23 \} }{ \{ 9 \} \{ 15 \} \{ 11 \} \{ 13 \} },&a_{3,4}(z) \equiv \dfrac{ \{ 5 \} \{ 19 \} \{ 1 \} \{ 23 \} }{ \{ 7 \} \{ 17 \} \{ 11 \} \{ 13 \} },\\&a_{1,3}(\sqrt{-1}z) \equiv \dfrac{ \{ 2 \} \{ 22 \} }{ \{ 10 \} \{ 14 \} },&a_{3,3}(z) \equiv \dfrac{ \{ 4 \} \{ 20 \} \{ 2 \} \{ 22 \} \{ 0 \} \{ 24 \} }{ \{ 8 \} \{ 16 \} \{ 10 \} \{ 14 \} \{ 12 \} ^2}. \end{aligned}$$

Since we have computed $d_{1,3}(z)$, $d_{2,3}(z)$ and $d_{3,4}(z)$,  (2.6) can be written as follows:

$$\begin{aligned}&\dfrac{ d_{1,3}(\sqrt{-1}(-q)^{-3}z)d_{3,3}((-q)^{1}z)}{d_{2,3}(\sqrt{-1}z)d_{3,4}(z)}\\&\quad = \dfrac{(z^2-q^2)(z^2-q^6)^{2}(z^2-q^{10})^{2 \text { or } 3}(z^2-q^{14})^{4}(z^2-q^{18})^3(z^2-q^{22})^2(z^2-q^{26})}{ (z^2-q^6)^2(z^2-q^{10})^3(z^2-q^{14})^4(z^2-q^{18})^3(z^2-q^{22})^2 } \end{aligned}$$

by (2.5) and

$$\begin{aligned}&\dfrac{ a_{2,3}(\sqrt{-1}z)a_{3,4}(z)}{ a_{1,3}(\sqrt{-1}(-q)^{-3}z)a_{3,3}((-q)^{1}z)} \\&\quad = \dfrac{ \{ 3 \} \{ 21 \} \{ 1 \} \{ 23 \} }{ \{ 9 \} \{ 15 \} \{ 11 \} \{ 13 \} } \times \dfrac{ \{ 5 \} \{ 19 \} \{ 1 \} \{ 23 \} }{ \{ 7 \} \{ 17 \} \{ 11 \} \{ 13 \} } \times \dfrac{ \{ 7 \} \{ 11 \} }{ \{ -1 \} \{ 19 \} } \times \dfrac{ \{ 9 \} \{ 17 \} \{ 11 \} \{ 15 \} \{ 13 \} ^2}{ \{ 5 \} \{ 21 \} \{ 3 \} \{ 23 \} \{ 1 \} \{ 25 \} } \\&\quad =\dfrac{ \{ 1 \} \{ 23 \} }{ \{ 25 \} \{ -1 \} } = \dfrac{(z^2-q^{-2})}{(z^2-q^{2})}. \end{aligned}$$

Thus we have

$$\begin{aligned}&\dfrac{(z^2-q^{-2})(z^2-q^6)^{2}(z^2-q^{10})^{2 \text { or } 3}(z^2-q^{14})^{4}(z^2-q^{18})^3(z^2-q^{22})^2(z^2-q^{26})}{ (z^2-q^6)^2(z^2-q^{10})^3(z^2-q^{14})^4(z^2-q^{18})^3(z^2-q^{22})^2 } \in \mathbf {k}[z^{\pm 1}], \end{aligned}$$

which implies our assertion. $\quad \square$

4 Types $E_7^{(1)}$ and $E_8^{(1)}$

In [2, App. C,D], there were several denominators which contain ambiguity for roots of higher order (note that there were several typos in [2, App. C,D]):

$$\begin{aligned} d^{E_7^{(1)}}_{3,4}(z)&=(z+q^{3})(z+q^{5})^2(z+q^{7})^2(z+q^{9})^{2+\epsilon ^{(1)}_1}(z+q^{11})^3(z+q^{13})^2(z+q^{15})^2(z+q^{17}), \nonumber \\ d^{E_7^{(1)}}_{4,5}(z)&=(z+q^{3})(z+q^{5})^2(z+q^{7})^{2+\epsilon ^{(1)}_1}(z+q^{9})^{2+\epsilon ^{(2)}_1}(z+q^{11})^{2+\epsilon }(z+q^{13})^3(z+q^{15})^2(z+q^{17}),\nonumber \\ d^{E_7^{(1)}}_{4,4}(z)&= (z-q^{2})(z-q^{4})^2(z-q^{6})^{2+\epsilon ^{(1)}_1}(z-q^{8})^{2+\epsilon ^{(1)}_2} \\&\quad \times (z-q^{10})^{2+\epsilon ^{(1)}_2}(z-q^{12})^{3+\epsilon ^{(1)}_1}(z-q^{14})^3(z-q^{16})^2(z-q^{18}), \nonumber \\ d^{E_8^{(1)}}_{2,4}(z)&= (z+q^{3})(z+q^{5})(z+q^{7})^{2}(z+q^{9})^{2}(z+q^{11})^{2+\epsilon _1^{(1)}}(z+q^{13})^{2+\epsilon _1^{(2)}} (z+q^{15})^{2+\epsilon _1^{(3)}}\nonumber \\&\quad \times (z+q^{17})^{2+\epsilon _1^{(2)}}(z+q^{19})^{2+\epsilon _1^{(1)}}(z+q^{21})^{3}(z+q^{23})^{2}(z+q^{25})^2(z+q^{27})(z+q^{29}), \nonumber \\ d^{E_8^{(1)}}_{3,3}(z)&=(z-q^{2})(z-q^{4})(z-q^{6})(z-q^{8})^2(z-q^{10})^2(z-q^{12})^{2+\epsilon _1^{(1)}}(z-q^{14})^3(z-q^{16})^2\\&\quad \times (z-q^{18})^{2+\epsilon _1^{(1)}}(z-q^{20})^3(z-q^{22})^2(z-q^{24})^2(z-q^{26})(z-q^{28})(z-q^{30}), \nonumber \\ d^{E_8^{(1)}}_{3,4}(z)&= (z+q^{3})(z+q^{5})^{2}(z+q^{7})^{2}(z+q^{9})^{2+\epsilon _1^{(1)}}(z+q^{11})^{2+\epsilon _2^{(1)}}(z+q^{13})^{2+\epsilon _2^{(2)}}(z+q^{15})^{2+\epsilon _2^{(3)}} \\&\quad \times (z+q^{17})^{2+\epsilon _2^{(2)}}(z+q^{19})^{2+\epsilon _2^{(1)}}(z+q^{21})^{3+\epsilon _1^{(1)}}(z+q^{23})^{3}(z+q^{25})^2(z+q^{27})^2(z+q^{29}), \nonumber \\ d^{E_8^{(1)}}_{3,5}(z)&=(z-q^{4})(z-q^{6})^{2}(z-q^{8})^{2}(z-q^{10})^{2+\epsilon _1^{(1)}}(z-q^{12})^{2+\epsilon _1^{(2)}}(z-q^{14})^{2+\epsilon _1^{(3)}}(z-q^{16})^{3+\epsilon _1^{(3)}}\\&\quad \times (z-q^{18})^{2+\epsilon _1^{(2)}}(z-q^{20})^{2+\epsilon _1^{(1)}}(z-q^{22})^{3}(z-q^{24})^{2}(z-q^{26})^2(z-q^{28}), \nonumber \\ d^{E_8^{(1)}}_{3,6}(z)&=\,(z+q^{5})(z+q^{7})^{2}(z+q^{9})^{2}(z+q^{11})^{2}(z+q^{13})^{2}(z+q^{15})^{2+\epsilon _1^{(1)}}(z+q^{17})^{3} \\&\quad \times (z+q^{19})^{2}(z+q^{21})^{2}(z+q^{23})^{2}(z+q^{25})^{2}(z+q^{27}), \nonumber \\ d^{E_8^{(1)}}_{4,4}(z)&= (z-q^{2})(z-q^{4})^{2}(z-q^{6})^{2+\epsilon _1^{(1)}}(z-q^{8})^{2+\epsilon _2^{(1)}}(z-q^{10})^{2+\epsilon _3^{(1)}}(z-q^{12})^{2+\epsilon _4^{(1)}}(z-q^{14})^{2+\epsilon _4^{(2)}}\\&\quad \times (z-q^{16})^{2+\epsilon _4^{(2)}}(z-q^{18})^{2+\epsilon _4^{(1)}}(z-q^{20})^{3+\epsilon _3^{(1)}}(z-q^{22})^{3+\epsilon _2^{(1)}}(z-q^{24})^{3+\epsilon _1^{(1)}}\\&\quad \times (z-q^{26})^3(z-q^{28})^2(z-q^{30}), \nonumber \\ d^{E_8^{(1)}}_{4,5}(z)&=(z+q^{3})(z+q^{5})^{2}(z+q^{7})^{2+\epsilon _1^{(1)}}(z+q^{9})^{2+\epsilon _2^{(1)}}(z+q^{11})^{2+\epsilon _2^{(2)}} (z+q^{13})^{2+\epsilon _3^{(1)}}(z+q^{15})^{2+\epsilon _3^{(2)}} \\&\quad \times (z+q^{17})^{2+\epsilon _3^{(1)}}(z+q^{19})^{3+\epsilon _2^{(2)}}(z+q^{21})^{2+\epsilon _2^{(1)}}(z+q^{23})^{3+\epsilon _1^{(1)}}(z+q^{25})^3(z+q^{27})^2(z+q^{29}), \nonumber \\ d^{E_8^{(1)}}_{4,6}(z)&= (z-q^{4})(z-q^{6})^{2}(z-q^{8})^{2+\epsilon _1^{(1)}}(z-q^{10})^{2+\epsilon _1^{(2)}}(z-q^{12})^{2+\epsilon _1^{(3)}}(z-q^{14})^{2+\epsilon _2^{(1)}} \\&\quad \times (z-q^{16})^{2+\epsilon _2^{(1)}}(z-q^{18})^{3+\epsilon _1^{(3)}}(z-q^{20})^{2+\epsilon _1^{(2)}}(z-q^{22})^{2+\epsilon _1^{(1)}}(z-q^{24})^{3}(z-q^{26})^2(z-q^{28}), \nonumber \\ d^{E_8^{(1)}}_{5,5}(z)&= (z-q^{2})(z-q^{4})(z-q^{6})^{2}(z-q^{8})^{2+\epsilon _1^{(1)}}(z-q^{10})^{2+\epsilon _1^{(2)}}(z-q^{12})^{2+\epsilon _2^{(1)}}(z-q^{14})^{2+\epsilon _2^{(2)}} \\&\quad \times (z-q^{16})^{2+\epsilon _2^{(2)}}(z-q^{18})^{2+\epsilon _2^{(1)}}(z-q^{20})^{3+\epsilon _1^{(3)}}(z-q^{22})^{2+\epsilon _1^{(1)}}(z-q^{24})^3(z-q^{26})^2(z-q^{28}), \nonumber \\ d^{E_8^{(1)}}_{5,6}(z)&=(z+q^{3})(z+q^{5})(z+q^{7})^2(z+q^{9})^2(z+q^{11})^{2+\epsilon _1^{(1)}}(z+q^{13})^{2+\epsilon _1^{(2)}}(z+q^{15})^{2+\epsilon _1^{(3)}}(z+q^{17})^{2+\epsilon _1^{(2)}} \\&\quad \times (z+q^{19})^{2+\epsilon _1^{(1)}}(z+q^{21})^3(z+q^{23})^2(z+q^{25})^2(z+q^{27})(z+q^{29}), \end{aligned}$$

for some $\epsilon _1^{(i)} \in \{0,1\}$, $\epsilon _2^{(i)} \in \{0,1,2\}$, $\epsilon _3^{(i)} \in \{0,1,2,3\}$ and $\epsilon _4^{(i)} \in \{0,1,2,3,4\}$$(i \in \mathbb {Z}_{\ge 0})$$(i \in \mathbb {Z}_{\ge 0})$.

Applying the same arguments in Sect. 2, we can prove that $\epsilon _k^{(i)} =k$ for all $1\le k \le 4$ and i. Hence there are roots of order 4, 5, 6 also. To the best knowledge of the authors, this is the first such observation of a root of order strictly larger than 3.

When we prove for $E_7^{(1)}$ cases, we employ the following homomorphisms

$$\begin{aligned}&V(3)_{(-q)^{-1}} \otimes V(6)_{(-q)^{4}} \twoheadrightarrow V(1)\otimes V(2)_{(-q)}\otimes V(7)_{(-q)^3}, \\&V(3)_{(-q)^{-1}} \otimes V(5)_{(-q)^{3}} \twoheadrightarrow V(1)\otimes V(2)_{(-q)}\otimes V(6)_{(-q)^2}, \\&V(1)_{(-q)^{-3}} \otimes V(4)_{(-q)^1} \twoheadrightarrow V(2)\otimes V(5), \end{aligned}$$

which can be obtained from Proposition 1.1 and Table 1.

Table 1 The minimal pairs $\underline{p}$ and their socles ${{\text {soc}}}_Q(\underline{p})$ of $\Gamma _Q$ in [2, App. A]