Quasi-symmetric function

quasi-symmetric polynomial (in combinatorics)

Let $X$ be a finite or infinite set (of variables) and consider the ring of polynomials $R[X]$ and the ring of power series $R[[X]]$ over a commutative ring $R$ with unit element in the commuting variables from $X$. A polynomial or power series $f(X) \in R[[X]]$ is called symmetric if for any two finite sequences of indeterminates $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_n$ from $X$ and any sequence of exponents $i_1,\ldots,i_n \in \mathbf{N}$, the coefficients in $f$ of $X_1^{i_1} \cdots X_n^{i_n}$ and $Y_1^{i_1} \cdots Y_n^{i_n}$ are the same.

Quasi-symmetric formal power series are a generalization introduced by I.M. Gessel, [a2], in connection with the combinatorics of plane partitions and descent sets of permutations [a3]. This time one takes a totally ordered set of indeterminates, e.g. $V = \{V_1,V_2,\ldots\}$, with the ordering that of the natural numbers, and the condition is that the coefficients of $X_1^{i_1} \cdots X_n^{i_n}$ and $Y_1^{i_1} \cdots Y_n^{i_n}$ are equal for all totally ordered sets of indeterminates $X_1 < \ldots < X_n$ and $Y_1 < \ldots < Y_n$. For example, $$ X_1 X_2^2 + X_1 X_3^2 + X_2 X_3^2 $$ is a quasi-symmetric polynomial in three variables that is not symmetric.

Products and sums of quasi-symmetric polynomials and power series are again quasi-symmetric (obviously), and thus one has, for example, the ring of quasi-symmetric power series $$ \widehat{ \mathbf{Q}^{\mathrm{sym}}_{\mathbf{Z}}(X)) } $$ in countably many commuting variables over the integers and its subring $$ \mathbf{Q}^{\mathrm{sym}}_{\mathbf{Z}}(X)) $$ of quasi-symmetric polynomials in finite of countably many indeterminates, which are the quasi-symmetric power series of bounded degree.

Given a word over , also called a composition in this context, consider the quasi-monomial function

defined by . These form a basis over the integers of .

The algebra of quasi-symmetric functions is dual to the Leibniz–Hopf algebra, or, equivalently to the Solomon descent algebra, more precisely, to the direct sum

of the Solomon descent algebras of the symmetric groups (cf. also Symmetric group), [a5], with a new multiplication over which the direct sum of the original multiplications is distributive. See [a1], [a4].

The algebra of quasi-symmetric functions in countably many indeterminates over the integers, , is a free polynomial algebra over the integers, [a6].

There is a completely different notion in the theory of functions of a complex variable that also goes by the name quasi-symmetric function; cf., e.g., [a7].

References

[a1]	I.M. Gel'fand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, "Noncommutative symmetric functions" Adv. Math. , 112 (1995) pp. 218–348
[a2]	I.M. Gessel, "Multipartite -partitions and inner product of skew Schur functions" Contemp. Math. , 34 (1984) pp. 289–301
[a3]	I.M. Gessel, Ch. Reutenauer, "Counting permutations with given cycle-structure and descent set" J. Combin. Th. A , 64 (1993) pp. 189–215
[a4]	C. Malvenuto, Ch. Reutenauer, "Duality between quasi-symmetric functions and the Solomon descent algebra" J. Algebra , 177 (1994) pp. 967–982
[a5]	L. Solomon, "A Mackey formula in the group ring of a Coxeter group" J. Algebra , 41 (1976) pp. 255–268
[a6]	M. Hazewinkel, "The algebra of quasi-symmetric functions is free over the integers" Preprint CWI (Amsterdam) and ICTP (Trieste) (1999)
[a7]	M. Chuaqui, B. Osgood, "Weak Schwarzians, bounded hyperbolic distortion, and smooth quasi-symmetric functions" J. d'Anal. Math. , 68 (1996) pp. 209–252