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Schubert polynomials and the nilCoxeter algebra. (English) Zbl 0809.05091
Schubert polynomials \({\mathfrak S}_ \sigma(x_ 1,x_ 2,\dots)\) indexed by permutations have been introduced and investigated by I. N. Bernstein, I. M. Gel’fand and S. I. Gel’fand [Russ. Math. Surveys 28, No. 3, 1-26 (1973; Zbl 0286.57025)], M. Demazure [Ann. Sci. École Norm. Sup., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)], and by A. Lascoux and M.-P. Schützenberger [C. R. Acad. Sci., Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)]; see also their paper [Symmetry and flag manifolds, Lect. Notes in Math. 996, 118-144 (1983; Zbl 0542.14031)].
In this paper the theory of Schubert polynomials is recovered using the nilCoxeter algebra \({\mathfrak C}_ n\) with the identity element \(e\), given by its generators and defining relations as the \(K\)-algebra \[ \begin{split} {\mathfrak C}_ n=\Bigl\langle u_ 1,\dots, u_{n-1}\mid u^ 2_ i= 0\;(i\in I_{n-1}),\;u_ i u_ j= u_ j u_ i\\ (\text{for }| i- j|\geq 2),\text{ and } u_ i u_{i+1} u_ i= u_{i+1} u_ i u_{i+1} (\text{for } i\in I_{n-2})\Bigr\rangle\end{split} \] over any commutative ring \(K\); here \(I_ n= \{1,2,\dots, n\}\). This algebra can be faithfully represented by the algebra of operators generated by \(\Phi_ i\) \((i\in I_{n-1})\), \[ \Phi_ i(\sigma)= \begin{cases} \sigma\tau_ i &\text{if } \ell(\sigma\tau_ i)= \ell(\sigma)+1;\\ 0 & \text{otherwise}.\end{cases} \] Here, \(\sigma\) is any permutation in the symmetric group \({\mathcal S}_ n\) defined on \(I_ n\), \(\tau_ i\) \((i\in I_{n-1})\) is the ‘adjacent’ transposition \((i,i+1)\), and \(\ell(\sigma)\) is the length of \(\sigma\in {\mathcal S}_ n\) defined as the minimal \(p\) such that \(\sigma= \tau_{a_ 1}\cdot\tau_{a_ 2}\cdot\dots\cdot \tau_{a_ p}\) for some \(a_ j\in I_{n-1}\). A sequence \(a= (a_ 1,\dots, a_ p)\), \(a_ j\in I_{n-1}\) is called a reduced decomposition of \(\sigma\) if \(p= \ell(\sigma)\). \(R(\sigma)\) denote the set of all reduced decompositions for \(\sigma\). For any reduced decomposition \(a= (a_ 1,\dots, a_ p)\) let us identify the monomial \(u_{a_ 1} u_{a_ 2}\cdots u_{a_ k}\) in \({\mathfrak C}_ n\) with \(\tau_{a_ 1} \cdot \tau_{a_ 2}\cdot\dots\cdot \tau_{a_ k}\) in \({\mathcal S}_ n\); the defining relations for \({\mathfrak C}_ n\) guarantee the correctness of such notation, and we see that \({\mathcal S}_ n\) gives a \(K\)-basis for \({\mathfrak C}_ n\). As usual, denote by \(\langle f,\sigma\rangle\) the coefficient of \(\sigma\in {\mathcal S}_ n\) in the \(K\)- expression for \(f\in {\mathfrak C}_ n\). Further, denote \[ A_ i(x)= (e+ xu_{n-1})\cdot (e+ xu_{n-2})\cdot\dots\cdot (e+ xu_ i) \] for any \(i\in I_{n-1}\), \(\bar x= (x_ 1,\dots, x_{n-1})\), \({\mathfrak S}(\bar x)= A_ 1(x_ 1)\cdot A_ 2(x_ 2)\cdot \dots\cdot A_{n-1}(x_{n- 1})\) and let \({\mathfrak S}_ \sigma(\bar x)= \langle{\mathfrak S}(\bar x),\sigma\rangle\). Among the results of this paper is Theorem 2.2 saying that \({\mathfrak S}_ \sigma(\bar x)\) is a Schubert polynomial. The authors prove also (Lemma 2.3) that in the case of \(\text{char } K= 0\), \[ {\mathfrak S}_ \sigma(1,\dots, 1)= {1\over p!} \sum_{(a_ 1,\dots, a_ p)\in R(\sigma)} a_ 1\cdots a_ p. \] Also proved is the \(q\)-analogue of this last formula conjectured by I. Macdonald [Notes on Schubert polynomials, LACIM, Université du Québec, Montréal (1991)]: \[ {\mathfrak S}_ \sigma(1,q,\dots, q^{n-2})= {1\over [1]\cdot[2]\cdot\dots\cdot [p]}\sum_{(a_ 1,\dots, a_ p)\in R(\sigma)} [a_ 1]\cdot\dots\cdot [a_ p]q^{\sum_{\{i\mid a_ i\leq a_{i+1}\}}} i, \] where \([t]= 1+ q+\cdots+ q^{t-1}\).

MSC:
05E05 Symmetric functions and generalizations
20C30 Representations of finite symmetric groups
14M15 Grassmannians, Schubert varieties, flag manifolds
05A19 Combinatorial identities, bijective combinatorics
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