non-commutative symmetric functions ii: combinatorics and...
TRANSCRIPT
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Non-commutative symmetric functions II:Combinatorics and coinvariants
Mike Zabrocki - York University
joint work with Nantel Bergeron, Anouk Bergeron-Brlek, Emmanuel Briand, Christophe Hohlweg, Christophe Reutenauer, Mercedes Rosas
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NCSym = Hopf algebra of set partitions
set partition of
e.g.
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n
Comparison Sym to NCSym
Sym NCSymn
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n
Comparison Sym to NCSym
Sym NCSymn
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n
Comparison Sym to NCSym
Sym NCSym
-invariants of
n
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n
Comparison Sym to NCSym
Sym NCSym
-invariants of -invariants of
n
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n
Comparison Sym to NCSym
Sym NCSym
-invariants of -invariants of
algebra of partitions
n
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n
Comparison Sym to NCSym
Sym NCSym
-invariants of -invariants of
algebra of partitions algebra of set partitions
n
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n
Comparison Sym to NCSym
Sym NCSym
-invariants of -invariants of
algebra of partitions algebra of set partitions
Linear span of sum of orbits of monomials
n
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Comparison Sym to NCSym
Sym has graded dimensions given by
NCSym has graded dimensions given by
1, 1, 2, 3, 5, 7, 11, ...
1, 1, 2, 5, 15, 52, 203, ...
n
n
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Harmonics of
the space of polynomials which are killed by allsymmetric differential operators
means replace variables in polynomial by differential operators
e.g.
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Properties of the harmonics of
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Properties of the harmonics of
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Properties of the harmonics of
Vandermonde determinant
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Properties of the harmonics of
Vandermonde determinant
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Properties of the harmonics of
Vandermonde determinant
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Properties of the harmonics of
Vandermonde determinant
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Coinvariants of
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Coinvariants of
Define a scalar product on
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Coinvariants of
Define a scalar product on
with respect to this scalar product
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Non-commutative setting
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Non-commutative setting
Choices!
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Non-commutative setting
Choices!
What is meant by non-commutative derivative?
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Non-commutative setting
Choices!
What is meant by non-commutative derivative?
Left ideal or two-sided ideal?
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A non-commutative derivative
Note in the commutative setting :
where
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Another non-commutative derivative
if
then
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Two types of non-commutative harmonics
the space of non-commutative polynomials which are killed by all symmetric differential operators
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Non-commutative coinvariants
Because of the duality between product and differentiation
we have that
where is the left ideal generated byelements of NCSym with no constant term
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More non-commutative coinvariants
But what about ?
shuffle is a commutative product on
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Properties of non-commutative harmonics
as modules NC-Chevalley like theorem
Proof idea:Exhibit explicit isomorphisms of LHS into
is realized by using the shuffle product
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Consequences of these isomorphisms
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Consequences of these isomorphisms
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Consequences of these isomorphisms
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Consequences of these isomorphisms
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Consequences of these isomorphisms
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Consequences of these isomorphisms
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Properties of non-commutative harmonics
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Non-commutative polynomials
universal enveloping algebraof free Lie algebra
free Lie algebra =
universal enveloping algebraof
by PBW theorem
can also show that this holds as modules
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Some interesting isomorphisms
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Some interesting isomorphisms
by Chevalley’s theorem
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Some interesting isomorphisms
by Chevalley’s theorem
but remember we showed already that
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Some interesting isomorphisms
by Chevalley’s theorem
but remember we showed already that
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Some interesting isomorphisms
by Chevalley’s theorem
but remember we showed already that
![Page 44: Non-commutative symmetric functions II: Combinatorics and ...garsia.math.yorku.ca/~zabrocki/talks/coinvariants.pdf · Non-commutative symmetric functions II: Combinatorics and coinvariants](https://reader036.vdocuments.us/reader036/viewer/2022081323/5f064a297e708231d4173f94/html5/thumbnails/44.jpg)
Some interesting isomorphisms
by Chevalley’s theorem
but remember we showed already that
![Page 45: Non-commutative symmetric functions II: Combinatorics and ...garsia.math.yorku.ca/~zabrocki/talks/coinvariants.pdf · Non-commutative symmetric functions II: Combinatorics and coinvariants](https://reader036.vdocuments.us/reader036/viewer/2022081323/5f064a297e708231d4173f94/html5/thumbnails/45.jpg)
Some open problems
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Some open problems
•Determine the coinvariants with respect the two sided idealis it even finite?
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Some open problems
•Determine the coinvariants with respect the two sided idealis it even finite?
•The analogous quotient of quasi-symmetric functionshas Catalan dimension. What happens in the non-commutative case?
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Some open problems
•Determine the coinvariants with respect the two sided idealis it even finite?
•The analogous quotient of quasi-symmetric functionshas Catalan dimension. What happens in the non-commutative case?
•General theory with other finite subgroups of