Non-commutative symmetric functions II: Combinatorics and...
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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
Transcript of Non-commutative symmetric functions II: Combinatorics and...
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
NCSym = Hopf algebra of set partitions
set partition of
e.g.
n
Comparison Sym to NCSym
Sym NCSymn
n
Comparison Sym to NCSym
Sym NCSymn
n
Comparison Sym to NCSym
Sym NCSym
-invariants of
n
n
Comparison Sym to NCSym
Sym NCSym
-invariants of -invariants of
n
n
Comparison Sym to NCSym
Sym NCSym
-invariants of -invariants of
algebra of partitions
n
n
Comparison Sym to NCSym
Sym NCSym
-invariants of -invariants of
algebra of partitions algebra of set partitions
n
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
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
Harmonics of
the space of polynomials which are killed by allsymmetric differential operators
means replace variables in polynomial by differential operators
e.g.
Properties of the harmonics of
Properties of the harmonics of
Properties of the harmonics of
Vandermonde determinant
Properties of the harmonics of
Vandermonde determinant
Properties of the harmonics of
Vandermonde determinant
Properties of the harmonics of
Vandermonde determinant
Coinvariants of
Coinvariants of
Define a scalar product on
Coinvariants of
Define a scalar product on
with respect to this scalar product
Non-commutative setting
Non-commutative setting
Choices!
Non-commutative setting
Choices!
What is meant by non-commutative derivative?
Non-commutative setting
Choices!
What is meant by non-commutative derivative?
Left ideal or two-sided ideal?
A non-commutative derivative
Note in the commutative setting :
where
Another non-commutative derivative
if
then
Two types of non-commutative harmonics
the space of non-commutative polynomials which are killed by all symmetric differential operators
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
More non-commutative coinvariants
But what about ?
shuffle is a commutative product on
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
Consequences of these isomorphisms
Consequences of these isomorphisms
Consequences of these isomorphisms
Consequences of these isomorphisms
Consequences of these isomorphisms
Consequences of these isomorphisms
Properties of non-commutative harmonics
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
Some interesting isomorphisms
Some interesting isomorphisms
by Chevalley’s theorem
Some interesting isomorphisms
by Chevalley’s theorem
but remember we showed already that
Some interesting isomorphisms
by Chevalley’s theorem
but remember we showed already that
Some interesting isomorphisms
by Chevalley’s theorem
but remember we showed already that
Some interesting isomorphisms
by Chevalley’s theorem
but remember we showed already that
Some open problems
Some open problems
•Determine the coinvariants with respect the two sided idealis it even finite?
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?
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
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