ON THE SLOPE OF THE SCHUR FUNCTOR OF A VECTOR BUNDLE

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1 International Journal of Pure and Applied Matheatics Volue 86 No , ISSN: (printed version; ISSN: (on-line version url: doi: PAijpa.eu ON THE SLOPE OF THE SCHUR FUNCTOR OF A VECTOR BUNDLE Elena Rubei Dipartiento di Mateatica e Inforatica U. Dini Viale Morgagni 67/A, 50134, Firenze, ITALY Abstract: We prove that, for any coplex vector bundle E of rank e on a copact Kähler anifold X, we have that µ(s λ E = λ µ(e for any λ = (λ 1,...,λ e 1 with λ i N and λ 1... λ e 1, where λ = λ λ e 1, the sybol S λ denotes the Schur functor and µ is the slope. This result has already been stated, without proof, by Ottaviani in AMS Subject Classification: 19L10, 55R10 Key Words: slope, Schur functors 1. Introduction In this short note we prove that for any coplex vector bundle E of rank e on a copact Kähler anifold X, we have that µ(s λ E = λ µ(e for any λ = (λ 1,...,λ e 1 with λ i N and λ 1... λ e 1, where λ = λ λ e 1, the sybol S λ denotes the Schur functor and µ is the slope. This resulthas already been stated, withoutproof, in [3] (and usedin the recent paper [4] In [1] it is stated and proved only in the case of the exterior powers. Received: March 20, 2013 c 2013 Acadeic Publications, Ltd. url:

2 522 E. Rubei Since the lack of a proof of the general case in the literature, we expose here a coplete proof. Aong the literature on the subject, we quote also [5]: in it, the author calculated the Chern characters of the syetric powers and of the exterior powers of a vector bundle. It sees difficult to generalize his results to any Schur functors of a vector bundle. 2. Notation and Recalls Let λ = (λ 1,...,λ e 1 with λ i N and λ 1... λ e 1. For any V coplex vector space of diension n, the sybol S λ V will denote the Schur representation (SL(V-representation associated to λ (see Lecture 6 in [2]. The S λ V are irreducible SL(V-representations and it is well-known that all the irreducible SL(V-representations are of this for. We recall that Pieri s forula says that, if ν = (ν 1,ν 2,... is a partition of a natural nuber d with ν 1 ν 2.. and t is a natural nuber, then S ν V S t V = γ Γ S γ V as SL(V-representation, where Γ is the set of all the partitions γ = (γ 1,γ 2,... with γ 1 γ 2... of d+t whose Young diagras are obtained fro the Young diagra of ν adding t boxes not two in the sae colun. Notation 1. If E is a coplex vector bundle on a copact Kähler anifold X, then µ(e will denote the slope of E, i.e. the degree of E divided by the rank of E. 3. The Proof Theore 2. For any coplex vector bundle E of rank e on a copact Kähler anifold X, we have that µ(s λ E = λ µ(e for any λ = (λ 1,...,λ e 1 with λ 1... λ e 1 0, where λ = λ λ e 1. Proof. First we prove the result in the case λ = (,0,...,0, i.e. in the case of the syetric powers of E. Obviously µ(s E = deg(s E rk(s E = deg(s E (, e+ 1

3 ON THE SLOPE OF so we have to prove that ( deg(s e+ 1 E = e deg(e. We prove it by induction on e. The case e = 1 is trivial. By the splitting principle, we can suppose that where rk(e i = 1 for i = 1,...,e. So E = E 1... E e, deg(s E = deg( i1,...,i e N, i i e= S i 1 E 1... S ie E e = deg(s i 1 E 1... S ie E e = i 1,...,i e N, i i e= i 1,...,i e N, i i e= = [ deg(s i 1 E deg(s ie E e ] i 1,...,i e N, i i e= [i 1 deg(e i e deg(e e ]. Observe that coefficients of deg(e 1,..., deg(e e in the above forula ust be equal. Besides the su of the coefficients of deg(e 1,..., deg(e e ust be i i e, i 1,...,i e N, i i e= that is ( e 1+. Therefore the coefficient of deg(ei for any i = 1,...,e ust ( e 1+. Thus we get be e deg(e = e ( e 1+ (deg(e deg(e e = e ( e 1+ deg(e, as we wanted to prove. Now we prove the result in general by induction on the nuber of the rows of the Young diagra of λ. If the nuber of the rows is 1, we already know the result. So suppose that the stateent holds for S α E with the nuber of the rows of α less or equal than k.

4 524 E. Rubei We want to prove the stateent when the nuber of the rows is less or equal than k +1; we show it by induction on the nuber t of the eleents of the (k +1-th row. If t = 0 we know the stateent by induction assuption. So suppose that the Young diagra of λ has k +1 rows and t eleents in the (k +1-th row. We define ν to be the Young diagra we get fro λ by deleting the last row. Consider Pieri s forula applied to S ν E S t E: S ν E S t E = γ Γ S γ E, where Γ is the set of all the partitions γ = (γ 1,γ 2,... with γ 1 γ 2... of ν + t whose Young diagras are obtained fro the Young diagra of ν adding t boxes not two in the sae colun. Obviuosly λ Γ, so we can write Γ = {λ} Γ. We have: therefore then, by induction assuption, µ(s ν E S t E = µ( γ Γ S γ E, µ(s ν E+µ(S t E = µ( γ Γ S γ E S λ E, ν µ(e+tµ(e = µ( γ Γ S γ E S λ E. Hence we get: Therefore ( ν +tµ(e = γ Γ deg(sγ E+deg(S λ E rk(s ν E S t. E rk(s ν E S t E ( ν +t µ(e = γ Γ deg(s γ E+deg(S λ E. Hence deg(s λ E = rk(s ν E S t E ( ν +t µ(e γ Γ deg(s γ E = rk(s ν E S t E ( ν +t µ(e γ Γ rk(s γ Eµ(S γ E = rk(s ν E S t E ( ν +t µ(e γ Γ rk(s γ E γ µ(e

5 ON THE SLOPE OF = µ(e ( ν +t rk(s ν E S t E rk(s γ E = µ(e ( ν +t rk(s λ E γ Γ = rk(s λ E λ µ(e, where the last but three equality holds by induction assuption (induction on t and the last equality and the last but two equality hold because λ = γ = ν +t. References [1] V. Ancona, G. Ottaviani, Stability of special instanton bundles on P 2n+1, Trans. Aer. Math. Soc., 341, No. 2 (1994, , doi: /S [2] W. Fulton, J. Harris, Representation Theory, A First Course Graduate Texts in Matheatics, USA, Springer Verlag (1991, doi: / [3] G. Ottaviani Varietà proiettive di codiensione piccola Quaderni INDAM, Aracne, Italy (1995, ISBN [4] E. Rubei, Stability of hoogeneous bundles on P 3, Geoetriae Dedicata, 158, No. 1 (2012, 1-21, doi: /s [5] D. Svrtan, New plethys operation, Chern characters of exterior and syetric powers with applications to Stiefel-Whitney classes of grassannians, Theoretical Coputer Science, 117 (1993, , doi: / ( S.

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