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Special bases for \(S_n\) and \(\text{GL}(n)\). (English) Zbl 0537.20006

The starting point of this paper is Springer’s construction of representations of Weyl groups [T. A. Springer, Invent. Math. 36, 173–207 (1976; Zbl 0374.20054); 44, 279-293 (1978; Zbl 0376.17002)]. For \(W=S_n\), the symmetric group, this construction gives a one-to-one correspondence between the set \(\widehat W\) of irreducible representations of \(W\) and the set of unipotent conjugacy classes in \(G=\text{GL}(n, \mathbb C)\). The authors use the topological version of Springer’s construction due to D. Kazhdan and G. Lusztig [Adv. Math. 38, 222–228 (1980; Zbl 0458.20035)] in which the representation \(\sigma (\Omega)\in \widehat W\) corresponding to a unipotent class \(\Omega\) is realized in the top homology (in the sense of Borel and Moore) of the variety \({\mathcal B}_ u\) of Borel subgroups of \(G\) containing a given element \(u\in \Omega\). This realization provides a special basis for \(\sigma (\Omega)\) formed by the homology classes of irreducible components of \({\mathcal B}_ u\); the authors call it the Springer basis. The first main result of the paper is that the Springer basis satisfies remarkable branching properties. The formulation requires some terminology.
Let \(P\) be a linear reductive algebraic group and suppose that the set \(\hat P\) of rational irreducible representations of \(P\) has the structure of a partially ordered set. For any \(P\)-module \(V\) and \(\sigma\in \hat P\) let \(V_{\sigma}\) denote the isotypic component of \(V\) of type \(\sigma\). Put \(V^{\sigma}=\sum_{\tau \leq \sigma}V_{\tau},\quad V^{\sigma}_{0}=\sum_{\tau<\sigma}V_{\tau}\) so that \(V^{\sigma}/V^{\sigma}_{0}=V_{\sigma}\). A basis \(B\) of \(V\) is said to be \(P\)-compatible if (a) each \(V^{\sigma}\) is spanned by some subset of \(B\), and (b) for any \(v\in(V^{\sigma}-V^{\sigma}_{0})\cap B\) the residue class \(v \bmod V^{\sigma}_{0}\) generates an irreducible \(P\)-module having as a basis the images in \(V^{\sigma}/V^{\sigma}_{0}\) of some vectors in \((V^{\sigma}-V^{\sigma}_{0})\cap B\). Now observe that \(\hat W\) has a natural partial ordering \((\sigma(\Omega_ 1)<\sigma(\Omega_ 2)\) iff \(\Omega_ 1\subset {\bar \Omega}_ 2)\); taking the product order we obtain a partial order on \(\hat P\) for any subgroup \(P\) of \(W\) of the form \(S_{n_ 1}\times...\times S_{n_ k}\).
Theorem (2.5). Let \(W=S_ n\) and \(V\) be an irreducible \(W\)-module. Then the Springer basis in \(V\) is \(P\)-compatible for any subgroup \(P\) of \(W\) of the form \(S_{n_ 1}\times...\times S_{n_ k}\).
The authors conjecture that the set of lines spanned by the vectors in the Springer basis is uniquely determined by this property. They claim that this can be proved for \(n\leq 6\). The authors also construct a special basis for any rational irreducible representation of \(G=\text{GL}(n)\). The construction uses the well-known relationship due to H. Weyl between irreducible representations of \(G\) and the irreducible representations of the groups \(S_ n\), and the Springer basis for \(S_ N\). This special basis is proved to be \(P\)-compatible for any subgroup \(P\) of \(G\) of the form \(\text{GL}(n_ 1)\times...\times \text{GL}(n_ k)\) (Theorem 3.4). The problem of a canonical definition of this basis seems to be very important.

MSC:

20C30 Representations of finite symmetric groups
20G05 Representation theory for linear algebraic groups
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References:

[1] Borel, A.; Moore, J. C., Homology theory for locally compact spaces, Michigan Math. J., 7, 137-153 (1960) · Zbl 0116.40301 · doi:10.1307/mmj/1028998385
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[6] Springer, T. A., Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math., 36, 173-204 (1976) · Zbl 0374.20054 · doi:10.1007/BF01390009
[7] Springer, T. A., A construction of representations of Weyl groups, Invent. Math., 44, 279-293 (1978) · Zbl 0376.17002 · doi:10.1007/BF01403165
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