Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.
(But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.
Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va
qVsoq have “weird” quantum dimensions.
The quantum dimension ofVso5
↑ q
Above: Kuperberg’sB2web relations∼1995. We wanted to
generalize Kuperberg’s results. We failed because quantization
is hard outside of typeA.
But let me explain what we can do. Using a coideal
subalgebra does the trick.
The action is constructed using
the unquantized diagrammatics.
Some delicate quantizations
Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.
(But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.
Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va
qVsoq have “weird” quantum dimensions.
The quantum dimension ofVso5
↑ q
Above: Kuperberg’sB2web relations∼1995. We wanted to
generalize Kuperberg’s results. We failed because quantization
is hard outside of typeA.
But let me explain what we can do. Using a coideal
subalgebra does the trick.
The action is constructed using
the unquantized diagrammatics.
Some delicate quantizations
Quantum skew Howe duality:
Lehrer–Zhang–Zhang∼2009.
(But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.
Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va
qVsoq have “weird” quantum dimensions.
The quantum dimension ofVso5
↑ q
Above: Kuperberg’sB2web relations∼1995. We wanted to
generalize Kuperberg’s results. We failed because quantization
is hard outside of typeA.
But let me explain what we can do. Using a coideal
subalgebra does the trick.
The action is constructed using
the unquantized diagrammatics.
Some delicate quantizations
Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.
(But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.
Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va
qVsoq have “weird” quantum dimensions.
The quantum dimension ofVso5
↑ q
Above: Kuperberg’sB2web relations∼1995. We wanted to
generalize Kuperberg’s results. We failed because quantization
is hard outside of typeA.
But let me explain what we can do. Using a coideal
subalgebra does the trick.
The action is constructed using
the unquantized diagrammatics.
Some delicate quantizations
Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.
(But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.
Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va
qVsoq have “weird” quantum dimensions.
The quantum dimension ofVso5
↑ q
Above: Kuperberg’sB2web relations∼1995. We wanted to
generalize Kuperberg’s results. We failed because quantization
is hard outside of typeA.
But let me explain what we can do. Using a coideal
subalgebra does the trick.
The action is constructed using
the unquantized diagrammatics.
Some delicate quantizations
Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.
(But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVglq is [n].
The quantum dimension ofVsoq is [n−1]+1.
Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va
qVsoq have “weird” quantum dimensions.
The quantum dimension ofVso5
↑ q
Above: Kuperberg’sB web relations∼1995.
We wanted to generalize Kuperberg’s
results. We failed because quantization
is hard outside of typeA.
But let me explain what we can do. Using a coideal
subalgebra does the trick.
The action is constructed using
the unquantized diagrammatics.
Some delicate quantizations
Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.
(But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVglq is [n].
The quantum dimension ofVsoq is [n−1]+1.
Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va
qVsoq have “weird” quantum dimensions.
The quantum dimension ofVso5
↑ q
Above: Kuperberg’sB web relations∼1995.
We wanted to generalize Kuperberg’s
results. We failed because quantization
is hard outside of typeA.
But let me explain what we can do.
Using a coideal subalgebra does the trick.
The action is constructed using
the unquantized diagrammatics.
Some delicate quantizations
Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.
(But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.
Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va
qVsoq have “weird” quantum dimensions.
The quantum dimension ofVso5
↑ q
Above: Kuperberg’sB2web relations∼1995. We wanted to
generalize Kuperberg’s results. We failed because quantization
is hard outside of typeA.
But let me explain what we can do.
Using a coideal subalgebra does the trick.
The action is constructed using
the unquantized diagrammatics.
Some delicate quantizations
Using aq-monoidal diagrammatic category Webq,qn we can define a full Howe functor ΦextBD such that we get a commuting diagram
U˙q(so2k) Rep0q(son)
Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.
(But its quite easy and not their main point.)
Does not quantize! Quantizes easily
No action at all. Action unclear.
The quantum dimension ofVglq is [n]. The quantum dimension ofVsoq is [n−1]+1.
Hence,Vsoq does not come fromVglq ! This “flaw” propagates all the way through:Va
qVsoq have “weird” quantum dimensions.
The quantum dimension ofVso5
↑ q
Above: Kuperberg’sB2web relations∼1995. We wanted to
generalize Kuperberg’s results. We failed because quantization
is hard outside of typeA.
But let me explain what we can do. Using a coideal
subalgebra does the trick.
The action is constructed using
the unquantized diagrammatics.