Symmetric functions in noncommuting variables

By Mercedes H. Rosas, Bruce E. Sagan. 2002

86 citations recorded by the source corpus.

Consider the algebra Q >of formal power series in countably many noncommuting variables over the rationals. The subalgebra Π(x 1 , x 2 ,...) 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, complete homogeneous, and Schur symmetric functions as well as investigating their properties.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4f3906a766aab1d7ffd15aa4a8916a47c7880d5f

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Citations in the corpus

Branching rules in the ring of superclass functions of unipotent upper-triangular matrices (2008, background): “The ring NCSym was introduced by Wolf [22], and further explored by Rosas and Sagan [19].”
Skew quasisymmetric Schur functions and noncommutative Schur functions (2010, methodology): “Then [52] defined the Schur function in noncommuting variables S λ to be S λ = ∑”
Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables (2005, methodology): “Define the vector space NCSymn = L{mA}A⊢[n] where here we have used mA as a symbol representing a basis element for A a set partition.”
Monoidal Functors, Species, and Hopf Algebras (2010, background): “The object Π is the Hopf algebra of symmetric functions in noncommuting variables studied in [106, 225, 276], Even though the Hopf monoid Π is commutative, the Hopf algebra Π is not.”
Commutative combinatorial Hopf algebras (2006, background): “s of σ. The sums (55) Uπ := X supp(σ)=π Mσ span a Hopf subalgebra ΠQSym of SQSym, which is isomorphic to the graded dual of the Hopfalgebra ofsymmetric functions in noncommuting variables (such as in [29, 4], not to be confused with Sym), which we will denote here by WSym(A), for Word symmetric func”
THE HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND QUASI-SYMMETRIC FUNCTIONS IN NON-COMMUTATIVE VARIABLES ARE FREE AND CO-FREE (2005, unknown): “C O ] 1 9”
Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras (2010, methodology): “A lucid introduction is given by Rosas and Sagan [35] and combinatorial applications by Gebhard and Sagan [19].”