27 MV cycles

27 MV cycles for $L (2 α_{1}^{\lor} + a_{2}^{\lor})$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 26 July 2013

λ^{\lor} = 2 (α_{1}^{\lor} + α_{2}^{\lor})

(1) $b_{1}^{00}$

\begin{matrix} p = b_{+}, wt = 0, dim = 0 \\ z_{b} = \overline{{t_{w_{0} λ^{\lor}} K}} \end{matrix}

(3) $b_{1}^{10}$

\begin{matrix} p = {\tilde{f}}_{1} b_{+}, wt = - α_{1}^{\lor}, dim = 1 \\ z_{b} = \overline{{x_{- α_{1} - δ} (d) t_{w_{0} λ^{\lor} + α_{1}^{\lor}} K | d \in ℂ^{\times}}} \end{matrix}

(6) $b_{1}^{01}$

\begin{matrix} p = {\tilde{f}}_{2} b_{+}, wt = - α_{2}^{\lor}, dim = 1 \\ z_{b} = \overline{{x_{- α_{2} - δ} (d) t_{w_{0} λ^{\lor} + α_{2}^{\lor}} K | d \in ℂ^{\times}}} \end{matrix}

(12) $b_{1}^{20}$

\begin{matrix} p = {\tilde{f}}_{1}^{2} b_{+}, wt = - 2 α_{1}^{\lor}, dim = 2 \\ z_{b} = \overline{{x_{- α_{1}} (d_{1}) x_{- α_{1} + e^{λ}} (d_{2}) t_{w_{0} λ^{\lor} + 2 α_{1}^{\lor}} K | d_{1} \in ℂ^{\times}, d_{2} \in ℂ}} \end{matrix}

(4) $b_{1}^{11}$

\begin{matrix} p = {\tilde{f}}_{2} {\tilde{f}}_{1} b_{+}, wt = - (α_{1}^{\lor} + α_{2}^{\lor}), dim = 2 \\ z_{b} = \overline{{x_{- α_{1} - δ} (d_{1}) x_{- α_{2} - 2 δ} (d_{2}) t_{w_{0} λ^{\lor} + α_{1}^{\lor} + α_{2}^{\lor}} K | d_{1}, d_{2} \in ℂ^{\times}}} \end{matrix}

(7) $b_{2}^{11}$

\begin{matrix} p = {\tilde{f}}_{1} {\tilde{f}}_{2} b_{+}, wt = - (α_{1}^{\lor} + α_{2}^{\lor}), dim = 2 \\ z_{b} = \overline{{x_{- α_{2} - δ} (d_{1}) x_{- α_{1} - 2 δ} (d_{2}) t_{w_{0} λ^{\lor} + α_{1}^{\lor} + α_{2}^{\lor}} K | d_{1}, d_{2} \in ℂ^{\times}}} \end{matrix}

(20) $b_{1}^{02}$

\begin{matrix} p = {\tilde{f}}_{2}^{2} b_{+}, wt = - 2 α_{2}^{\lor}, dim = 2 \\ z_{b} = \overline{{x_{- α_{2}} (d_{1}) x_{- α_{2} + δ} (d_{2}) t_{w_{0} λ^{\lor} + 2 α_{2}^{\lor}} K | d_{1} \in ℂ^{\times}, d_{2} \in ℂ}} \end{matrix}

(13) $b_{1}^{21}$

\begin{matrix} p = {\tilde{f}}_{2} {\tilde{f}}_{1}^{2} b_{+} = {\tilde{f}}_{1} {\tilde{f}}_{2} {\tilde{f}}_{1} b_{+}, wt = - 2 α_{1}^{\lor} - α_{2}^{\lor}, dim = 3 \\ z_{b} = \overline{{x_{- α_{1}} (d_{1}) x_{- α_{1} + δ} (d_{2}) x_{- α_{2} - 3 δ} (d_{3}) t_{w_{0} λ^{\lor} - 2 α_{1}^{\lor} - α_{2}^{\lor}} K | d_{1}, d_{3} \in ℂ^{\times}, d_{2} \in ℂ}} \end{matrix}

(9) $b_{2}^{21}$

\begin{matrix} p = {\tilde{f}}_{1}^{2} {\tilde{f}}_{2} b_{+}, wt = - 2 α_{1}^{\lor} - α_{2}^{\lor}, dim = 3 \\ z_{b} = \overline{{x_{- α_{2} 0 δ} (d_{1}) x_{α_{1} - δ} (d_{2}) x_{- α_{1}} (d_{3}) t_{w_{0} λ^{\lor} + 2 α_{1}^{\lor} + α_{2}^{\lor}} K | d_{1}, d_{2} \in ℂ^{\times}, d_{3} \in ℂ}} \end{matrix}

(21) $b_{1}^{12}$

\begin{matrix} p = {\tilde{f}}_{1} {\tilde{f}}_{2}^{2} b_{+}, wt = - α_{1}^{\lor} - 2 α_{2}^{\lor}, dim = 3 \\ z_{b} = \overline{{x_{- α_{2}} (d_{1}) x_{- α_{2} + δ} (d_{2}) x_{- α_{1} - 3 δ} (d_{3}) t_{w_{0} λ^{\lor} + α_{1}^{\lor} + 2 α_{2}^{\lor}} K | d_{1}, d_{3} \in ℂ^{\times}, d_{2} \in ℂ}} \end{matrix}

(8) $b_{2}^{12}$

\begin{matrix} p = {\tilde{f}}_{2}^{2} {\tilde{f}}_{1} b_{+} = {\tilde{f}}_{2} {\tilde{f}}_{1} {\tilde{f}}_{2} b_{+}, wt = - α_{1}^{\lor} - 2 α_{2}^{\lor}, dim = 3 \\ z_{b} = \overline{{x_{- α_{2} - δ} (d_{1}) x_{- φ - 2 δ} (d_{2}) x_{- α_{2}} (d_{3}) t_{w_{0} λ^{\lor} + α_{1}^{\lor} + 2 α_{2}^{\lor}} K | d_{1}, d_{2} \in ℂ^{\times}, d_{3} \in ℂ}} \end{matrix}

(15) $b_{2}^{31}$

\begin{matrix} p = {\tilde{f}}_{1}^{3} {\tilde{f}}_{2} b_{+}, wt = - 3 α_{1}^{\lor} - α_{2}^{\lor}, dim = 4 \\ z_{b} = \overline{{x_{- α_{1}} (d_{1}) x_{- φ - δ} (d_{2}) x_{- α_{1} + δ} (d_{3}) x_{- α_{1} + δ} (d_{4}) t_{w_{0} λ^{\lor} + 3 α_{1}^{\lor} + α_{2}^{\lor}} K | d_{1}, d_{2} \in ℂ^{\times}, d_{3}, d_{4} \in ℂ}} \end{matrix}

(14) $b_{1}^{22}$

\begin{matrix} p = {\tilde{f}}_{2}^{2} {\tilde{f}}_{1}^{2} b_{+}, wt = - 2 α_{1}^{\lor} - 2 α_{2}^{\lor}, dim = 4 \\ z_{b} = \overline{{x_{- α_{1}} (d_{1}) x_{- α_{2} - 2 δ} (d_{2}) x_{- φ - δ} (d_{3}) x_{- α_{2} - δ} (d_{4}) t_{w_{0} λ^{\lor} + 2 α_{1}^{\lor} + 2 α_{2}^{\lor}} K | d_{1}, d_{2} \in ℂ^{\times}, d_{3}, d_{4} \in ℂ}} \end{matrix}

(10) $b_{2}^{22}$

\begin{matrix} p = {\tilde{f}}_{1} {\tilde{f}}_{2}^{2} {\tilde{f}}_{1} b_{+} = {\tilde{f}}_{2} {\tilde{f}}_{1}^{2} {\tilde{f}}_{2} b_{+}, wt = - 2 α_{1}^{\lor} - 2 α_{2}^{\lor}, dim = 4 \\ z_{b} = \overline{{x_{- α_{2} - δ} (d_{1}) x_{- α_{1} - δ} (d_{2}) x_{- α_{2} - δ} (d_{3}) x_{- φ - δ} (d_{4}) t_{w_{0} λ^{\lor} + 2 α_{1}^{\lor} + 2 α_{2}^{\lor}} K | d_{1}, d_{2}, d_{3} \in ℂ^{\times}, d_{4} \in ℂ}} \end{matrix}

(23) $b_{3}^{22}$

\begin{matrix} p = {\tilde{f}}_{1}^{2} {\tilde{f}}_{2}^{2} b_{+}, wt = - 2 α_{1}^{\lor} - 2 α_{2}^{\lor}, dim = 4 \\ z_{b} = \overline{{x_{- α_{2}} (d_{1}) x_{- α_{2} + δ} (d_{2}) x_{- α_{1} - 2 δ} (d_{3}) x_{- α_{1} - δ} (d_{4}) t_{w_{0} λ^{\lor} + 2 α_{1}^{\lor} + 2 α_{2}^{\lor}} K | d_{1}, d_{3} \in ℂ^{\times}, d_{2}, d_{4} \in ℂ}} \end{matrix}

(22) $b_{1}^{13}$

\begin{matrix} p = {\tilde{f}}_{2}^{3} {\tilde{f}}_{1} b_{+}, wt = - α_{1}^{\lor} - 3 α_{2}^{\lor}, dim = 4 \\ z_{b} = \overline{{x_{- α_{2}} (d_{1}) x_{- α_{2} + δ} (d_{2}) x_{- φ - δ} (d_{3}) x_{- α_{2} + 2 δ} (d_{4}) t_{w_{0} λ^{\lor} + α_{1}^{\lor} + 3 α_{2}^{\lor}} K | d_{1}, d_{3} \in ℂ^{\times}, d_{2}, d_{4} \in ℂ}} \end{matrix}

(16) $b_{2}^{32}$

\begin{matrix} p = {\tilde{f}}_{2} {\tilde{f}}_{1}^{3} {\tilde{f}}_{2} b_{+} = {\tilde{f}}_{1}^{2} {\tilde{f}}_{2}^{2} {\tilde{f}}_{1} b_{+}, wt = - 3 α_{1}^{\lor} - 2 α_{2}^{\lor}, dim = 5 \\ z_{b} = \overline{{x_{- α_{1}} (d_{1}) x_{- φ - δ} (d_{2}) x_{- α_{1} + δ} (d_{3}) x_{- α_{2} - 2 δ} (d_{4}) x_{- φ} (d_{5}) t_{w_{0} λ^{\lor} + 3 α_{1}^{\lor} + 2 α_{2}} K | d_{1}, d_{2}, d_{4} \in ℂ^{\times}, d_{3}, d_{5} \in ℂ}} \end{matrix}

(25) $b_{3}^{32}$

\begin{matrix} p = {\tilde{f}}_{1}^{3} {\tilde{f}}_{2}^{2} b_{+}, wt = - 3 α_{1}^{\lor} - 2 α_{2}^{\lor}, dim = 5 \\ z_{b} = \overline{{x_{- α_{2}} (d_{1}) x_{- α_{1} - δ} (d_{2}) x_{- φ} (d_{3}) x_{- α_{1}} (d_{4}) x_{- α_{2} + δ} (d_{5}) t_{w_{0} λ^{\lor} + 3 α_{1}^{\lor} + 2 α_{2}^{\lor}} K | d_{1}, d_{2} \in ℂ^{\times}, d_{3}, d_{4}, d_{5} \in ℂ}} \end{matrix}

(24) $b_{2}^{23}$

\begin{matrix} p = {\tilde{f}}_{2}^{2} {\tilde{f}}_{1}^{2} {\tilde{f}}_{2} b_{+} = {\tilde{f}}_{1} {\tilde{f}}_{2}^{3} {\tilde{f}}_{1} b_{+}, wt = - 2 α_{1}^{\lor} - 3 α_{2}^{\lor}, dim = 5 \\ z_{b} = \overline{{x_{- α_{2}} (d_{1}) x_{- α_{2} + δ} (d_{2}) x_{- α_{1} - 2 δ} (d_{3}) x_{- α_{2} + δ} (d_{4}) x_{- φ} (d_{5}) t_{w_{0} λ^{\lor} + 2 α_{1}^{\lor} + 3 α_{2}^{\lor}} K | d_{1}, d_{3}, d_{4} \in ℂ^{\times}, d_{2}, d_{5} \in ℂ}} \end{matrix}

(17) $b_{1}^{23}$

\begin{matrix} p = {\tilde{f}}_{2}^{3} {\tilde{f}}_{1}^{2} b_{+}, wt = - 2 α_{1}^{\lor} - 3 α_{2}^{\lor}, dim = 5 \\ z_{b} = \overline{{x_{- α_{1}} (d_{1}) x_{- α_{2} - δ} (d_{2}) x_{- α_{2}} (d_{3}) x_{- φ} (d_{4}) x_{- α_{2} + δ} (d_{5}) t_{w_{0} λ^{\lor} + 2 α_{1}^{\lor} + 3 α_{2}^{\lor}} K | d_{1}, d_{2} \in ℂ^{\times}, d_{3}, d_{4}, d_{5} \in ℂ}} \end{matrix}

(29) $b_{3}^{42}$

\begin{matrix} p = {\tilde{f}}_{1}^{4} {\tilde{f}}_{2}^{2} b_{+}, wt = - 4 α_{1}^{\lor} - 2 α_{2}^{\lor}, dim = 6 \\ z_{b} = \overline{{x_{- α_{2}} (d_{1}) x_{- α_{1}} (d_{2}) x_{- α_{1} + δ} (d_{3}) x_{- φ + δ} (d_{4}) x_{- α_{1} + 2 δ} (d_{5}) x_{- α_{1} + 3 δ} (d_{6}) t_{w_{0} λ^{\lor} + 4 α_{1}^{\lor} + 2 α_{2}^{\lor}} | \binom{d_{1}, d_{2} \in ℂ^{\times},}{d_{3}, d_{4}, d_{5}, d_{6} \in ℂ}}} \end{matrix}

(18) $b_{2}^{33}$

\begin{matrix} p = {\tilde{f}}_{1}^{1} {\tilde{f}}_{2}^{3} {\tilde{f}}_{1}^{2} b_{+} = {\tilde{f}}_{2}^{2} {\tilde{f}}_{1}^{3} {\tilde{f}}_{2}^{1} b_{+}, wt = - 3 α_{1}^{\lor} - 3 α_{2}^{\lor}, dim = 6 \\ z_{b} = \overline{{x_{- α_{1}} (d_{1}) x_{- α_{2} - δ} (d_{2}) x_{- α_{1}} (d_{3}) x_{- φ} (d_{4}) x_{- α_{2}} (d_{5}) x_{- φ + δ} (d_{6}) t_{w_{0} λ^{\lor} + 3 α_{1}^{\lor} + 3 α_{2}^{\lor}} K | \binom{d_{1}, d_{2}, d_{3} \in ℂ^{\times},}{d_{4}, d_{5}, d_{6} \in ℂ}}} \end{matrix}

(26) $b_{3}^{33}$

\begin{matrix} p = {\tilde{f}}_{1}^{2} {\tilde{f}}_{2}^{3} {\tilde{f}}_{1} b_{+} = {\tilde{f}}_{2}^{1} {\tilde{f}}_{1}^{3} {\tilde{f}}_{2}^{2} b_{+}, wt = - 3 α_{1} - 3 α_{2}, dim = 6 \\ z_{b} = \overline{{x_{- α_{2}} (d_{1}) x_{- α_{1} - δ} (d_{2}) x_{- φ} (d_{3}) x_{- α_{1}} (d_{4}) x_{- α_{2}} (d_{5}) x_{- φ + δ} (d_{6}) t_{w_{0} λ^{\lor} + 3 α_{1}^{\lor} + 3 α_{2}^{\lor}} K | \binom{d_{1}, d_{2}, d_{5} \in ℂ^{\times},}{d_{3}, d_{4}, d_{6} \in ℂ}}} \end{matrix}

(27) $b_{1}^{24}$

\begin{matrix} p = {\tilde{f}}_{2}^{4} {\tilde{f}}_{1}^{2} b_{+}, wt = - 2 α_{1}^{\lor} - 4 α_{2}^{\lor}, dim = 6 \\ z_{b} = \overline{{x_{- α_{2}} (d_{1}) x_{- φ} (d_{2}) x_{- α_{2} + δ} (d_{3}) x_{- α_{2} + 2 δ} (d_{4}) x_{- φ + δ} (d_{5}) x_{- α_{2} + 3 δ} (d_{6}) t_{w_{0} λ^{\lor} + 2 α_{1}^{\lor} + 4 α_{2}^{\lor}} K | \binom{d_{1}, d_{2} \in ℂ^{\times},}{d_{3}, d_{4}, d_{5}, d_{6} \in ℂ}}} \end{matrix}

(30) $b_{3}^{43}$

\begin{matrix} p = {\tilde{f}}_{2} {\tilde{f}}_{1}^{4} {\tilde{f}}_{2}^{2} b_{+} = {\tilde{f}}_{1}^{3} {\tilde{f}}_{2}^{3} {\tilde{f}}_{1} b_{+}, wt = - 4 α_{1}^{\lor} - 3 α_{2}^{\lor}, dim = 7 \\ z_{b} = \overline{{x_{- α_{2}} (d_{1}) x_{- α_{1}} (d_{2}) x_{- α_{1} + δ} (d_{3}) x_{- φ + δ} (d_{4}) x_{- α_{1} + 2 δ} (d_{5}) x_{- α_{2} - δ} (d_{6}) x_{- φ + 2 δ} (d_{7}) | \binom{d_{1}, d_{2}, d_{6} \in ℂ^{\times},}{d_{3}, d_{4}, d_{5}, d_{7} \in ℂ}}} \end{matrix}

(28) $b_{2}^{34}$

\begin{matrix} p = {\tilde{f}}_{2}^{3} {\tilde{f}}_{1}^{3} {\tilde{f}}_{2} b_{+} = {\tilde{f}}_{1} {\tilde{f}}_{2}^{4} {\tilde{f}}_{1}^{2} b_{+}, wt = - 3 α_{1}^{\lor} - 4 α_{2}^{\lor}, dim = 7 \\ z_{b} = \overline{{\binom{x_{- α_{2}} (d_{1}) x_{- φ} (d_{2}) x_{- α_{2} + δ} (d_{3}) x_{- α_{1}} (d_{4}) x_{- φ + δ} (d_{5}) x_{- α_{2} + 2 δ} (d_{6})}{\cdot x_{- φ + 2 δ} (d_{7}) t_{w_{0} λ^{\lor} + 3 α_{1}^{\lor} + 4 α_{2}^{\lor}} K} | \binom{d_{1}, d_{2}, d_{4} \in ℂ^{\times},}{d_{3}, d_{5}, d_{6}, d_{7} \in ℂ}}} \end{matrix}

(31) $b_{3}^{44}$

\begin{matrix} p = {\tilde{f}}_{2}^{2} {\tilde{f}}_{1}^{4} {\tilde{f}}_{2}^{2} b_{+} = {\tilde{f}}_{1}^{2} {\tilde{f}}_{2}^{4} {\tilde{f}}_{1}^{2} b_{+}, wt = - 4 α_{1}^{\lor} - 4 α_{2}^{\lor}, dim = 8 \\ z_{b} = \overline{{\binom{x_{- α_{2}} (d_{1}) x_{- α_{1}} (d_{2}) x_{- α_{2}} (d_{3}) x_{- φ + δ} (d_{4}) x_{- α_{1} + δ} (d_{5}) x_{- φ + 2 δ} (d_{6})}{\cdot x_{- α_{2} + δ} (d_{7}) x_{- φ + 3 δ} (d_{8}) t_{w_{0} λ^{\lor} + 4 α_{1}^{\lor} + 4 α_{2}^{\lor}} K} | \binom{d_{1}, d_{2}, d_{3} \in ℂ^{\times},}{d_{4}, d_{5}, d_{6}, d_{7}, d_{8} \in ℂ}}} \end{matrix}

page history

27 MV cycles for L(2α1∨+a2∨)

(1) b100

(3) b110

(6) b101

(12) b120

(4) b111

(7) b211

(20) b102

(13) b121

(9) b221

(21) b112

(8) b212

(15) b231

(14) b122

(10) b222

(23) b322

(22) b113

(16) b232

(25) b332

(24) b223

(17) b123

(29) b342

(18) b233

(26) b333

(27) b124

(30) b343

(28) b234

(31) b344

27 MV cycles for $L (2 α_{1}^{\lor} + a_{2}^{\lor})$

(1) $b_{1}^{00}$

(3) $b_{1}^{10}$

(6) $b_{1}^{01}$

(12) $b_{1}^{20}$

(4) $b_{1}^{11}$

(7) $b_{2}^{11}$

(20) $b_{1}^{02}$

(13) $b_{1}^{21}$

(9) $b_{2}^{21}$

(21) $b_{1}^{12}$

(8) $b_{2}^{12}$

(15) $b_{2}^{31}$

(14) $b_{1}^{22}$

(10) $b_{2}^{22}$

(23) $b_{3}^{22}$

(22) $b_{1}^{13}$

(16) $b_{2}^{32}$

(25) $b_{3}^{32}$

(24) $b_{2}^{23}$

(17) $b_{1}^{23}$

(29) $b_{3}^{42}$

(18) $b_{2}^{33}$

(26) $b_{3}^{33}$

(27) $b_{1}^{24}$

(30) $b_{3}^{43}$

(28) $b_{2}^{34}$

(31) $b_{3}^{44}$