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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)]: ${\mathfrak S}_ \sigma(1,q,\dots, q^{n-2})= {1\over \cdot\cdot\dots\cdot [p]}\sum_{(a_ 1,\dots, a_ p)\in R(\sigma)} [a_ 1]\cdot\dots\cdot [a_ p]q^{\sum_{\{i\mid a_ i\leq a_{i+1}\}}} i,$ 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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