Quasi-symmetric functions, noncommutative symmetric functions, and Hecke algebras at \(q=0\). (Fonctions quasi-symétriques, fonctions symétriques non commutatives et algèbres de Hecke à \(q=0\).) (French. Abridged English version) Zbl 0839.20017
In the classical theory of Frobenius, the ring \(R=\bigoplus_{n\geq 0} R(\Sigma_n)\) of integral linear combinations of irreducible representations of the symmetric groups \(\Sigma_n\) over \(\mathbb{C}\) can be identified with the ring \(Sym\) of symmetric functions in a sequence of variables \(x_1, x_2, \dots\) by means of the characteristic map, which sends the Specht module \(V_\lambda\) to the Schur function \(s_\lambda\).
The authors give an analogue of this theory for representations of the Hecke algebra \(H_n (0)\) of type \(A\) over \(\mathbb{C}\). The irreducible representations \(C_I\) of \(H_n (0)\) are all 1-dimensional, and are indexed by the \(2^{n-1}\) subsets of \(\{1, 2,\dots, n-1\}\), or equivalently by the \(2^{n-1}\) compositions \(I\) of \(n\). Let \({\mathcal G}=\bigoplus_{n\geq 0} G_0 (H_n (0))\) be the ring of integral linear combinations of irreducible representations of the algebras \(H_n (0)\), and let \({\mathcal K}=\bigoplus_{n\geq 0} K_0 (H_n(0))\) be the analogous ring constructed from the projective covers \(P_I\) of the irreducible modules \(C_I\). The authors construct generalized characteristic maps \({\mathcal F}: {\mathcal G}\to QSym\), \({\mathcal E}:{\mathcal K}\to \mathbf {Sym}\), where \(QSym\) is the ring of quasi-symmetric functions in the \(x_i\), introduced by Gessel in 1984, and \(\mathbf {Sym}\) is the ring of non-commutative symmetric functions in the \(x_i\), introduced by the present authors and others in 1995. (See I. M. Gelfand et al. [Adv. Math. 112, No. 2, 218-348 (1995; Zbl 0831.05063)].) In fact, \(\mathbf {Sym}\) is the graded dual of the Hopf algebra \(QSym\), reflecting the duality between the groups \(G_0 (H_n (0))\) and \(K_0 (H_n (0))\).
To define \({\mathcal F}\) and \({\mathcal E}\), we require certain functions \(F_I\) in the \(x_i\), called quasi-ribbons, which form an additive basis of \(QSym\). The quasi-symmetric function \(F_I\) is an analogue of the skew Schur function associated with the ribbon, or difference of Young diagrams, indexed by the composition \(I\). It is defined as the (formal) sum of the monomials \(x^{j_1}_{a_1} \dots x^{j_m}_{a_m}\), where \(a_1< \dots< a_m\) and \(J=(j_1, \dots, j_m)\) refines \(I\). In particular, \(F_{(n)}\) and \(F_{(1^n)}\) are analogues of the complete symmetric function \(h_n\) and the elementary symmetric function \(e_n\) respectively. The map \({\mathcal F}\) sends \(C_I\) to \(F_I\), and the map \({\mathcal E}\) sends \(P_I\) to \(R_I\), the dual of \(F_I\) with respect to the quasi-ribbon basis of \(QSym\). As well as obtaining related character and induction formulae, it is shown that the natural map from \({\mathcal G}\) to \({\mathcal K}\) which sends \(P_I\) to \(C_I\) corresponds via the characteristic maps \({\mathcal F}\) and \({\mathcal E}\) to the map \(\mathbf{Sym}\to Sym\subset QSym\) which sends \(R_{(n)}\) to \(h_n\).
The author conclude by remarking that their results suggest that there should be an appropriate definition of quasi-symmetric functions associated to a root system, and mention that such a construction exists in unpublished work of Gessel.
The authors give an analogue of this theory for representations of the Hecke algebra \(H_n (0)\) of type \(A\) over \(\mathbb{C}\). The irreducible representations \(C_I\) of \(H_n (0)\) are all 1-dimensional, and are indexed by the \(2^{n-1}\) subsets of \(\{1, 2,\dots, n-1\}\), or equivalently by the \(2^{n-1}\) compositions \(I\) of \(n\). Let \({\mathcal G}=\bigoplus_{n\geq 0} G_0 (H_n (0))\) be the ring of integral linear combinations of irreducible representations of the algebras \(H_n (0)\), and let \({\mathcal K}=\bigoplus_{n\geq 0} K_0 (H_n(0))\) be the analogous ring constructed from the projective covers \(P_I\) of the irreducible modules \(C_I\). The authors construct generalized characteristic maps \({\mathcal F}: {\mathcal G}\to QSym\), \({\mathcal E}:{\mathcal K}\to \mathbf {Sym}\), where \(QSym\) is the ring of quasi-symmetric functions in the \(x_i\), introduced by Gessel in 1984, and \(\mathbf {Sym}\) is the ring of non-commutative symmetric functions in the \(x_i\), introduced by the present authors and others in 1995. (See I. M. Gelfand et al. [Adv. Math. 112, No. 2, 218-348 (1995; Zbl 0831.05063)].) In fact, \(\mathbf {Sym}\) is the graded dual of the Hopf algebra \(QSym\), reflecting the duality between the groups \(G_0 (H_n (0))\) and \(K_0 (H_n (0))\).
To define \({\mathcal F}\) and \({\mathcal E}\), we require certain functions \(F_I\) in the \(x_i\), called quasi-ribbons, which form an additive basis of \(QSym\). The quasi-symmetric function \(F_I\) is an analogue of the skew Schur function associated with the ribbon, or difference of Young diagrams, indexed by the composition \(I\). It is defined as the (formal) sum of the monomials \(x^{j_1}_{a_1} \dots x^{j_m}_{a_m}\), where \(a_1< \dots< a_m\) and \(J=(j_1, \dots, j_m)\) refines \(I\). In particular, \(F_{(n)}\) and \(F_{(1^n)}\) are analogues of the complete symmetric function \(h_n\) and the elementary symmetric function \(e_n\) respectively. The map \({\mathcal F}\) sends \(C_I\) to \(F_I\), and the map \({\mathcal E}\) sends \(P_I\) to \(R_I\), the dual of \(F_I\) with respect to the quasi-ribbon basis of \(QSym\). As well as obtaining related character and induction formulae, it is shown that the natural map from \({\mathcal G}\) to \({\mathcal K}\) which sends \(P_I\) to \(C_I\) corresponds via the characteristic maps \({\mathcal F}\) and \({\mathcal E}\) to the map \(\mathbf{Sym}\to Sym\subset QSym\) which sends \(R_{(n)}\) to \(h_n\).
The author conclude by remarking that their results suggest that there should be an appropriate definition of quasi-symmetric functions associated to a root system, and mention that such a construction exists in unpublished work of Gessel.
Reviewer: G.Walker (Manchester)
