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 $w=[a_1,\ldots,a_n]$ over $\mathbf{N}$, also called a composition in this context, consider the quasi-monomial function $$ M_w = \sum_{Y_1 < \cdots < Y_n} Y_1^{a_1}\cdots Y_n^{a_n} $$ defined by $w$. These form a basis over the integers of $\mathbf{Q}^{\mathrm{sym}}_{\mathbf{Z}}(X))$.

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 $$ \mathcal{D} = \bigoplus_n D(S_n) $$ of the Solomon descent algebras $D(S_n)$ 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, $\mathbf{Q}^{\mathrm{sym}}_{\mathbf{Z}}(X))$, 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] and Quasi-symmetric function of a complex variable.

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 $P$-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