A new tableau representation for supersymmetric Schur functions
The so-called supersymmetric skew Schur functions \(s_{\lambda/ \mu}\) in variables \(X_m=(x_1, x_2, \dots, x_m)\) and \(Y_n=(y_1, y_2, \dots, y_n)\) is related to the corresponding symmetric function in \(X_m\cup Y_n\) by the involutory automorphism on the ring of symmetric polynomials in \(Y_n\): \[ s_{\lambda/ \mu} (X_m/ Y_n)=\sum_{\mu\subseteq \nu\subseteq \lambda} s_{\nu/ \mu} (X_m) s_{\lambda'/ \nu'} (Y_n), \] have combinatorial interest in their own right, beyond their wide application, for example in the character theory of Lie superalgebras [\textit{V. G. Kac}, Adv. Math. 26, 8-96 (1977; Zbl 0366.17012)], in the formalism of the KP and derived integrable hierarchies, and related algebraic-geometrical contexts. The above formula for the case \(\mu=\emptyset\) was given by \textit{P. H. Dondi} and \textit{P. D. Jarvis} [Z. Phys. C 4, 201-205 (1980), J. Phys. A 14, 547-563 (1981; Zbl 0449.17002)], studied by \textit{A. B. Balantekin} and \textit{I. Bars} [in J. Math. Phys. 22, 1149-1162, 1810-1818 (1981; Zbl 0469.22017, Zbl 0547.22014)], formulated for the skew case by \textit{A. Berele} and \textit{A. Regev} [Adv. Math. 64, 118-175 (1987; Zbl 0617.17002)]\ and further developed by \textit{R. C. King} [Ars Comb. 16A, 269-287 (1983; Zbl 0547.22016)]. The authors present a new tableau description of such functions, and consequently derive and shed light on several other combinatorial characterizations and related (super)symmetric functions. Let \({\mathcal T}_{\lambda/\mu} (A)\) be the set of skew tableaux of shape \(\lambda/\mu\) with entries in \(A\subseteq \mathbb{Z}\), \(T(\alpha)\) the entry of cell \(\alpha\) belonging to some \(T\in {\mathcal T}_{\lambda, \mu} (A)\), and \(C(\alpha)\) the content of that cell (where \(C(i,j)=j-i)\). Then the major theorem of the paper reads \[ s_{\lambda/\mu} (X/Y)=\sum_{T\in {\mathcal T}_{\lambda/ \mu} (\mathbb{Z})} \prod_{\alpha\in \lambda/\mu} (x_{T (\alpha)}+ y_{T(\alpha)+ C(\alpha)}) \] for supersymmetric skew Schur functions of integer-indexed sets of indeterminates \(X\), \(Y\). (This theorem has also been obtained independently by \textit{I. G. MacDonald} [Actes 28e Seminaire Lotharingien, Publ. IRMA Strasbourg, 498/S27, 5-39 (1992)]\ using a different approach.) The authors' proof uses a modification of the Gessel-Viennot lattice path construction for the weight function \((x_j+ y_{i+j})\) on horizontal steps \((i,j)\to (i+1, j)\), to show that the determinantal form generated by elementary one-part partitions is indeed the involutive transform of the standard Jacobi-Trudi form of Schur functions in terms of complete functions. As an outcome of the analysis, a direct combinatorial proof is provided for the symmetry (in \(X\)) of the tableau form using a modified Bender-Knuth algorithm on certain `circled' tableaux. This enables the cancellation and the Berele-Regev factorization properties of standard supersymmetric functions to be established directly. A final extension of the work is to the \textit{L. C. Biedenharn} and \textit{J. D. Louck} factorial symmetric functions [Adv. Appl. Math. 9, No. 4, 447-464 (1988; Zbl 0707.33003), Proc. Natl. Acad. Sci. USA 87, No. 4, 1441-1445 (1990; Zbl 0711.33013)], which are realized as particular specializations of supersymmetric Schur functions for suitably (finitely) indexed variables. In this connection it is also possible to introduce a separate class of `factorial supersymmetric functions' \(s_{\lambda/\mu} (X_m/ Y_n|A_p)\) explicitly in a further finite set of parameters \((a_1, a_2, a_3, \dots, a_p)\), as in \textit{A. Molev} [in `Topics in Representation Theory II', Am. Math. Soc. (to appear)], and for which standard determinantal, factorization, Sergeev-Pragacz, and other results can be established.
scientific article; zbMATH DE number 703243