We study a family of ideals that were first introduced by De Concini and Procesi in their study of the flag variety, and that reveals to have an important role also in other contexts. These ideals of the polynomial ring are indexed by partitions of n. When the indexing partition is a hook we find a minimal set of generators for the De Concini-Procesi ideal, and we compute explicit formulas for its free minimal resolution.
We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of studied in Invariants and coinvariants of the symmetric group in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis.
We study the Hopf algebra structure for the symmetric functions in noncommutative variables studied in Symmetric functions in noncommuting variables, , and show how they are related to other well studied hopf algebras like the algebra of symmetric functions and the algebra of noncommutative symmetric functions of Gelfand, Krob, Lascoux, Leclerc, Retakh, and Thibon. We also find analogs of Chevalley's theorem stating that the ring of polynomials in noncommutative variables is isomorphic to the tensor product of its invariants times its coinvariants.
We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes.
Professor François Bergeron wrote some very nice slides explaining our results to an audience of general mathematicians at Dartmouth.
Consider the algebra of formal power series in countably many noncommuting variables over the rationals. The subalgebra \Pi(x) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, and complete homogeneous, and Schur symmetric functions as well as investigating their properties.
El profesor Bruce Sagan escribió una serie de láminas muy interesantes que tratan acerca de las funciones simétricas no conmutativas y problemas de coloración de grafos Graph coloring and symmetric functions in noncommuting variables.
A MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action of the symmetric group. In this article, we give a combinatorial overview of the Hopf algebra structure of the MacMahon symmetric function relying on the construction of a Hopf algebra from any alphabet of neutral letters due to Gian-Carlo Rota and Joel Stein.
A MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action of the symmetric group. We use a combinatorial construction of the different bases of the vector space of MacMahon symmetric functions found by the author to obtain their image under the principal specialization: the powers, risings and falling factorials. Then, we compute the connection coefficients of the different polynomial bases in a combinatorial way.
A MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action
of the symmetric group. In this article, we show that the MacMahon symmetric functions are the generating functions for the
orbits of sets of functions indexed by partitions under the diagonal action of a Young subgroup of a symmetric group.
We define a MacMahon chromatic symmetric function that generalizes Stanley's chromatic symmetric function. Then, we study
some of the properties of this new function thru its connection with the noncommutative chromatic symmetric
function of Gebhard and Sagan.
The Kronecker product of two Schur functions, is the Frobenius
characteristic of the tensor product of the irreducible
representations of the symmetric group. The coefficients of the
Kronecker product of two Schur functions in the Schur basis are
the multiplicity of the irreducible representations of the
symmetric group is such a tensor product. They are called
the Kronecker coefficients.
We use Sergeev's Formula for a Schur
function of a difference of two alphabets and the
comultiplication expansion for a Schur function to find
closed formulas for the Kronecker coefficients corresponding
hook shapes or two-row shapes.
We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes.
Consider the algebra of formal power series in countably many noncommuting variables over the rationals. The subalgebra \Pi(x) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, and complete homogeneous, and Schur symmetric functions as well as investigating their properties.
The Kronecker product of two Schur functions, is the Frobenius
characteristic of the tensor product of the irreducible
representations of the symmetric group. The coefficients of the
Kronecker product of two Schur functions in the Schur basis are
the multiplicity of the irreducible representations of the
symmetric group is such a tensor product. They are called
the Kronecker coefficients.
We use Sergeev's Formula for a Schur
function of a difference of two alphabets and the
comultiplication expansion for a Schur function to find
closed formulas for the Kronecker coefficients corresponding
hook shapes or two-row shapes.
A MacMahon symmetric function is a formal power series
in a finite number of alphabets that is invariant under
the diagonal action
of the symmetric group. In this article, we show that
the MacMahon symmetric functions are the generating
functions for the orbits of sets of functions indexed
by partitions under the diagonal action of a Young subgroup
of a symmetric group.
We define a MacMahon chromatic symmetric function that
generalizes Stanley's chromatic symmetric function. Then,
we study some of the properties of this new function
thru its connection with the noncommutative chromatic symmetric
function of Gebhard and Sagan.
Escrita bajo la dirección de Ira Gessel.