PBW Bases

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

Last updates: 30 September 2010

PBW bases for $𝔤 = 𝔰 𝔩_{2}$

In type

S L_{2}

there is only one reduced word for the longest element

w_{0} = s_{1}

, the only positive root is

α_{1}

and the negative root vector is

F_{α_{1}} = F_{1} .

PBW bases for $𝔤 = 𝔰 𝔩_{3}$

In type

A_{2}

with reduced word

w_{0} = s_{1} s_{2} s_{1}

the roots are

β_{1} = α_{1}, β_{2} = s_{1} (α_{2}) = α_{1} + α_{2} = ρ, β_{3} = s_{1} s_{2} (α_{1}) = α_{2},

and the negative root vectors are

F_{α_{1}} = F_{1}, F_{ρ} = T_{1} (F_{2}) = - F_{2} F_{1} + q F_{1} F_{2}, F_{α_{2}} = T_{1} T_{2} (F_{1}) = F_{2} .

For the reduced word

w_{0} = s_{2} s_{1} s_{2}

the root vectors are

β_{1} = α_{1}, β_{2} = s_{2} (α_{1}) = α_{1} + α_{2} = ρ, β_{3} = s_{2} s_{1} (α_{2}) = α_{1},

and the negative roots are

F_{α_{2}} = F_{2}, F_{ρ} = T_{1} (F_{1}) = - F_{1} F_{2} + q F_{2} F_{2}, F_{α_{1}} = T_{2} T_{1} (F_{2}) = F_{1} .

The canonical basis is the set

E_{1}^{(m)} E_{2}^{(m')} E_{1}^{(m'')} with m' \geq m + m'',

which satisfies

E_{1}^{(m)} E_{2}^{(m + m'')} E_{1}^{(m'')} = E_{2}^{(m'')} E_{1}^{(m + m'')} E_{2}^{(m)} .

m_{1} \leq m_{2}

the space

𝕌_{- m_{1} α_{1} - m_{2} α_{2}}

has bases

\begin{array}{ccc} w_{0} = s_{1} s_{2} s_{1} & canonical basis & w_{0} = s_{2} s_{1} s_{2} \\ F_{1}^{(m_{1})} F_{2}^{(m_{2})} & F_{1}^{(m_{1})} F_{2}^{(m_{2})} & F_{2}^{(m_{2} - m_{1})} F_{ρ}^{(m_{1})} \\ F_{1}^{(m_{1} - 1)} F_{ρ} F_{2}^{(m_{2} - 1)} & F_{1}^{(m_{1} - 1)} F_{2}^{(m_{2})} F_{1} & F_{2}^{(m_{2} - m_{1} + 1)} F_{ρ}^{(m_{1} - 1)} F_{1} \\ F_{1}^{(m_{1} - 2)} F_{ρ}^{(2)} F_{2}^{(m_{2} - 2)} & F_{1}^{(m_{1} - 2)} F_{2}^{(m_{2})} F_{1}^{(2)} & F_{2}^{(m_{2} - m_{1} + 2)} F_{ρ}^{(m_{1} - 2)} F_{1}^{(2)} \\ \dots & \dots & \dots \\ F_{ρ}^{(m_{1})} F_{2}^{(m_{2} - m_{1})} & F_{2}^{(m_{2})} F_{1}^{(m_{1})} & F_{2}^{(m_{2})} F_{1}^{(m_{1})} \end{array}

with the canonical basis in the middle and the two PBW bases on each side. If

m_{1} \geq m_{2}

, the space

𝕌_{- m_{1} α_{1} - m_{2} α_{2}}

has bases

\begin{array}{ccc} w_{0} = s_{1} s_{2} s_{1} & canonical basis & w_{0} = s_{2} s_{1} s_{2} \\ F_{1}^{(m_{1})} F_{2}^{(m_{2})} & F_{1}^{(m_{1})} F_{2}^{(m_{2})} & F_{ρ}^{(m_{2})} F_{1}^{(m_{1} - m_{2})} \\ F_{1}^{(m_{1} - 1)} F_{ρ} F_{2}^{(m_{2} - 1)} & F_{2} F_{1}^{(m_{1})} F_{2}^{(m_{2} - 1)} & F_{2} F_{ρ}^{(m_{2} - 1)} F_{2}^{(m_{1} - m_{2} + 1)} \\ F_{1}^{(m_{1} - 2)} F_{ρ}^{(2)} F_{2}^{(m_{2} - 2)} & F_{2}^{(2)} F_{1}^{(m_{1})} F_{2}^{(m_{2} - 2)} & F_{2}^{(2)} F_{ρ}^{(m_{2} - 2)} F_{2}^{(m_{1} - m_{2} + 2)} \\ \dots & \dots & \dots \\ F_{1}^{(m_{1} - m_{2})} F_{ρ}^{(m_{2})} & F_{2}^{(m_{2})} F_{1}^{(m_{1})} & F_{2}^{(m_{2})} F_{1}^{(m_{1})} \end{array}

with the canonical basis in the middle and the two PBW bases on each side. The most compact way to write the three bases is interms of paths in the corresponding MV polytope. Comparing the bases shows that the conversion between the indexings of the two PBW bases is

R_{s_{1} s_{2} s_{1}}^{s_{2} s_{1} s_{2}} (t_{1}, t_{2}, t_{3}) = \{\begin{array}{ll} (t_{2} + t_{3} - t_{1}, t_{1}, t_{2}), & if t_{1} \leq t_{3}, \\ (t_{2}, t_{3}, t_{1} + t_{2} - t_{3}), & if t_{1} \geq t_{3} . \end{array}

Notes and references

The canonical basis is given on p. 447 of [Lu].

References

[Lu] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447--498. MR1035415 (90m:17023)

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.

page history

PBW Bases

PBW bases for 𝔤=𝔰𝔩2

PBW bases for 𝔤=𝔰𝔩3

Notes and references

References

PBW bases for $𝔤 = 𝔰 𝔩_{2}$

PBW bases for $𝔤 = 𝔰 𝔩_{3}$