Schur-Weyl duality

LetNdenote ann-dimensional vector space overkwith group of linear automorphisms denoted by GL N. The r-fold tensor productN ^r has a natural structure of GL N -modules. ButN ^r is also a representation of the symmetric group, withσ Σr acting by permutation

σ v1 vr vσ1 vσr

The actions of GL N andΣrcommute. Therefore we obtain ring homomorphisms

kΣr End_GLn N ^r (8.1)

kGLN EndΣ_r N ^r (8.2)

The first map is the subject of this chapter. It is an isomorphism whenr n, see (Carter and Lusztig 1974); for larger values of nthe map is no longer injective, but it remains surjective, see (de Concini and Procesi 1976). In the first three sections of the chapter we give an account of these facts.

The surjectivity of (8.1) allow us to consider EndGLn N ^r -modules as representa-tions of the symmetric group. By ring theory, there is one irreducible representation of EndGLn N ^r for each isomorphism class of indecomposable summands inN ^r. The dimension of the irreducible module is given by the multiplicity of the indecomposable module inN ^r.

Our interest in the centralizer property (8.1) derives from the fact thatN ^ris a tilting module. Thus, every indecomposable summand is tilting. We want to know the multiplicity of each indecomposable tilting module inN ^r, as this multiplicity is equal to the dimen-sion of an irreducible representation of the symmetric group. The multiplicity formula of Theorem 6.12 allow us to count the multiplicities of some of the indecomposable tilting modules, thus to calculate the dimension of some of the simple modules of the symmetric group. This application of Theorem 6.12 is the subject of this chapter.

Notation and recollections

We identify GL N with the group GLn of invertiblen nmatrices with en-tries fromk, by choosing the natural basis ofN kⁿ.

The set of diagonal matrices is a maximal torus with character group denoted byX.

We letεidenote projection on the ii ’th entry of a matrix; thenε1 εnis a -basis ofX. Let be the bilinear form that makesε1 εnan orthonormal basis.

InsideX we have the root system Rof typeAn 1; the roots are αij εi

εj i j^! and we may choose αi αii^" 1 i 1 n 1^! as the set of simple roots.

For a root systems of typeAthe co-roots identify with the rootsα^#$ αfor all roots. Letωi ε1^%&'% εi. Then ωiα^#_j δij, andω1 ωnare a second

-basis ofX; these weights are called fundamental weights.

For each rootα we have a root subgroupUα. Then GLn is generated by all root subgroups together with the maximal torus. Let Bdenote the Borel

69

70 8. SCHUR-WEYL DUALITY

subgroup generated by the maximal torus and all root subgroups corresponding to negative roots. LetU denote the group generated by the root subgroups corresponding to positive roots.

The finite Weyl groupW0is generated bys1sn 1, wheresimapsε_itoε_{i 1}, εi 1toεi, and fixes allεjwith j ii 1. This gives an explicit isomorphism to the symmetric group onnletters.

Lets0denote the mapλ λλα₁

nα pα . The affine Weyl group is the group generated bys0s1sn. This group usually acts onXby the dot action wλ wλ ρ ρ, whereρis the sum of all fundamentals weights∑iωi.

A weightn1ε1 nnεnis called polynomial when allni 0. We letPn denote the set of polynomial and dominant weights. Expressed in both bases of Xthis set is

Pn n1ε1 nnεn n1 nn 0

m1ω1 mnωn mi 0 for alli

The degree of a polynomial weightn1ε1 nnεnis the sum∑ini. Let us verify that, as promised,

LEMMA8.1. N ^ris a tilting module.

P^ROOF. The weights ofNareε1εn. These weights comprise one Weyl group orbit. Since the set of weights of a module is preserved by the finite Weyl group,Nis a simple module with highest weightε1. This also shows thatNis the simple quotient of the Weyl moduleVnε1 Vnω1. But this Weyl module is simple becauseω1is minimal among dominant weights. This proves thatNis a simple Weyl module, hence tilting. Now

the lemma follows from Theorem 2.15. ^!

A partition ofris a sequence of non-negative non-increasing integersn1 nn 0 with sumr. The following map defines a one-to-one correspondence between the set of dominant polynomial weights of GLn and the set of partitions with at mostnnon-zero parts:

n1ε1 nnεn^" n1 nn 0

Hence we may think of an element ofPn as a partition as well as a polynomial dominant weight. The same map defines a one-to-one correspondence between the set of dominant polynomial GLn -weights of degreerand the set of partitions ofrwith at mostnnon-zero parts.

Recall thatn1 nn 0 isp singularif there is a sequence ofpequal consecutive partsni ni p. A partition isp-regular if it is notp-singular.

Recall from (James 1978) the definition of a family ofΣr-representations, the Specht-modules Sp^λparametrized by partitions ofr. In prime characteristic these modules are not necessarily simple (as they are in characteristic zero); but the Specht-modules parametrized byp-regular partitions have simple head, which we denote byD^λ. Then D^λ λap-regular partition ofr is a full set of non-isomorphic simple representations of the symmetric group Σr.

Restriction fromGLn toGLn 1

The proof of surjectivity of (8.1) requires some knowledge about restrictions to GLn 1. This section provides the necessary tools. We hasten to add that these results really be-long in the much broader context of restrictions to Levi subgroups, see (Donkin 1993).

Here, however, we do not need the full strength of of the results in (Donkin 1993).

We embed GLn 1 in GLn, and think of GLn 1 as sitting in the top left corner.

# GLn 1

1 ^$&% GLn

RESTRICTION FROM GLN TO GLN 1 71

To be more precise, we consider the subgroup of GLn generated by kerεnand the root subgroupsUαijwithi j n. For a GLn-moduleMwe writeMGLn 1 for the restriction to the subgroup GLn 1. The character group of kerεnis the free -module with bases ε1εn 1andω1ωn 1(We should really write “ε1restricted to kerεn” etc.).

NOTATION 8.2. In this chapter we will consider GLn-modules and GLn 1 -modules. To avoid confusion we denote the Weyl, induced, and tilting GLn-modules with highest weightλby Vnλ, H_n⁰λ, and Tnλ.

Let us consider the restriction ofVnωi (i n) to GLn 1. Sinceωi is minimal among dominant weights, it follows that

chVnωi

∑

wω_iw W₀

ewωi

Asεnrestricts to the identity we find that

chVnωi GLn 1 chVn 1ωi chVn 1ωi 1

By minimality of the fundamental weights among dominant weights both modules on the right hand side are simple. The GLn 1-weightsωi andωi 1 does not belong to the same orbit under the affine Weyl group; so the linkage principle ensures that

Vnωi GLn 1 Vn 1ωi Vn 1ωi 1

We see that the restriction of the Weyl moduleVnωi has a filtration of Weyl modules. In this section we will show that this holds for all Weyl modules of GLn.

THEOREM8.3. Suppose that M isGLn-module with a Weyl filtration (resp. a good filtration). Then the restriction M_GLn 1 has a Weyl-filtration (resp. a good filtration). If M is tilting, then M_GLn 1 is tilting.

We will prove the statement for induced modules; the corresponding statement for modules with a Weyl-filtration is then immediate as the dual module has a good filtration.

The statement about restriction of a tilting module is clear from the first part of the theorem.

Before we begin to prove the theorem, note the following lemma. Recall thatH⁰λ

Ind^G_Bkλ and that induction is left exact; we let (as is usual)Hⁱ denote the derived functors.

LEMMA8.4. Let M be a finite dimensional B-module. Assume that for each weight, µ, of M we have H_n¹µ 0. Then H_n⁰M has a good filtration, and H_n¹M 0

P^ROOF. Induction in the dimension ofM.

PROOF OFTHEOREM8.3. It is enough to show that the modulesH_n⁰λ,λ X^! re-stricts to a module with a good filtration by Corollary 2.9.

Consider now a dominant weightλ, soλ n1ω1#""" nnωnwith eachni ^$ 0. The case∑ini 0 is trivial. When∑ini 1, we have already proved the result in the beginning of this section as the Weyl modulesVnωi are simple, hence isomorphic toH_n⁰ωi. We will proceed inductively, so choose a fundamental weightωso thatλ ωis dominant.

From the surjectiveB-module homomorphismp:H_n⁰ω^&%'λ ω⁽⁾ ω^*λ ω we obtain a long exact sequence beginning with

0 ⁾ H_n⁰kerp⁺⁾ H_n⁰H_n⁰ω^%*λ ω^,) H_n⁰λ⁺⁾ H_n¹kerp The weights of kerpare all on the formλ wω,wω ω. By Kempfs vanishing theorem we haveH_n¹λ wω 0 asλis dominant and^-wωα^{.0/1 32} 0⁵⁴ 1⁶ in typeAn 1. Thus, by Lemma 8.4, we see thatH_n¹kerp 0 and thatH_n⁰kerp has a good filtration. We have (using the tensor identity)

0 ⁾ H_n⁰kerp⁺⁾ H_n⁰ω^% H_n⁰λ ω⁷⁾ H_n⁰λ⁺⁾ 0 (8.3) We see that each good factorH_n⁰ν ofH_n⁰kerp has highest weightν⁸ λ; so by induction each H_n⁰ν restricts to a module with a good filtration. Then Corollary 2.9 assures that

72 8. SCHUR-WEYL DUALITY

H_n⁰ kerp_GLn 1 has a good filtration. Also by induction,H_n⁰ λ ω_GLn 1 has a good filtration. SoH_n⁰ω H_n⁰ λ ω is a tensor product of two GL n 1-modules with good filtrations, hence has a good filtration. By Corollary 2.9 we conclude that the restriction of

H_n⁰ λ has a good filtration.

REMARK8.5. The Weyl factors in Vn λ_GLn 1 are known. See (Brundan, Kleshchev and Suprunenko 1998, Proposition A.2) for an explicit description.

R^EMARK8.6. The inductive argument in the proof of Theorem 8.3 may also be used to give a reasonable short proof that H⁰ λ H⁰ µ has a good filtration. This approach then yields a proof of the stability of the family of tilting modules under tensor products in type A.

The functorTr^r Given a GL n-moduleMwe note that

M^r

∑

λ; degλ r

Mλ

is a submodule as each root subgroupUαpreserve the degree. In factM^r is a summand of M as the sum of the weights spacesM_λwith deg λ rare preserved by the affine Weyl group; the linkage principle states that two simple modules may extend non-trivially only when their highest weights are in the same orbit of the affine Weyl group. We say thatM is a GL n-module in (homogeneous) degreerwhenM^r M. There are no homomor-phisms between modules in unequal degree, since GL n-linear homomorphisms preserve the degree of a module.

We will now define a functor from the category of GL n -modules to the category of GL n 1-modules of degreer.

Tr^rM M^rr_GLn 1

This a basically the functor Tr^nε1 used in (Donkin 1993). The functor is exact as it is the composition of truncations to a summand and a restriction, and these are exact. We have, as a first propery of Tr^r,

P^ROPOSITION8.7. If M has a Weyl or a good filtration, then so doesTr^rM. Hence Tr^rtakes tilting modules to tilting modules.

PROOF. Summands in modules with a good filtration has a good filtration, see Corol-lary 2.9. And Theorem 8.3 shows that the restriction of such a module to GL n 1 has a

good filtration.

Recall the definition of the set P n of polynomial dominant GL n-weights. We considerP n 1 as a subset ofP n byn1ε1 nn 1εn 1 n1ε1 nn 1εn 1

0εn. Compare the following proposition with the example given in the beginning of the previous section. The proof of the proposition is a simplified version of a proof sketched in (Donkin 1983).

PROPOSITION8.8. Letλ P n have degree r.

(i) Ifλ P n 1 thenTr^rVn λ Vn 1 λ, (ii) Ifλ P n 1 thenTr^rVn λ 0.

PROOF. Assume thatλ P n 1. We will prove that chTr^rVnλ chVn 1 λ

This will show (i) as Theorem 8.3 guarantees that Tr^rVn λ has a Weyl filtration. The equality of characters is established by checking that the dimension of each weight space agree. The character of the Weyl module is independent of the characteristic of the ground

THE FUNCTOR Tr^R 73

field, and we may thus apply our characteristic zero methods. We use Kostants formula for the dimension of a weight space:

dimVn λµ

∑

wW₀

1^lwp µ wλ (8.4)

Herep ν denotes the number of waysνcan be written as a sum of negative roots.

Letµbe a weight ofVn λ withµ wλ 0. We may assume thatµ P n 1 and that deg µ r, otherwise theµ-weight spaces of Tr^rVn λ andVn 1 λ are zero. Note that a weight 0 hasεn-coefficient 0, and that a weight 0 hasεn-coefficient 0.

Fromµ wλ 0 andλ wλ 0 we find that theεn-coefficient ofwλ is zero, since λµ P n 1. Sincewλ ρ ρhasεn-coefficient equal to zero, it follows thatw W0

fixesεn. It follows that in (8.4) we need only sum overwthat fixesεn.

We consider the weights of GL n 1 as a subspace of GL n-weight space. On this space the Weyl group of GL n 1 may be identified with the subgroup of the GL n-Weyl group, that fixesεn. Further, asµ wλ P n 1 we may just as well calculatep µ wλ as a weight of GL n 1. We have

dimTr^rVn λµ

∑

w W₀GLn

1^lwp µ wλ

∑

w W₀GLn 1

1^lw p µ wλ dimVn 1 λ_µ

This shows (i). To see (ii) we note thatλ P n 1 means that theεn-coefficient is 0.

Thenλ µshows that theεn-coefficient ofµis 0. So we see thatµ P n 1. It follows that theµ-weight space ofVn λ is killed by Tr^r. We are done.

REMARK 8.9. There is a similar statement about induced modules; the proof for Proposition 8.8 above is essentially a character calculation - and the characters of Vn λ and H_n⁰ λ are equal.

We prepare the proof of Theorem 8.11 below with a lemma.

LEMMA8.10. Letλ, µ P r. Then

Tr^r: Hom_GLn Vn λH_n⁰ µ Hom_GLn 1 Tr^rVn λTr^rH_n⁰ µ is surjective.

PROOF. We are done if Tr^rVn λ or Tr^rH_n⁰ µ is zero. So we may as well assume that λ,µ P n 1 . Further Hom_GLn 1 Tr^rVn λTr^rH_n⁰ µ 0 ifλ µ. So we need only considerµ λ. Now let f :Vn λ H_n⁰ λ be a non-zero GL n-homomorphism. Then Tr^rf : Tr^rVn λ Tr^rH_n⁰ λ is non-zero on theλ-weight spaces, which are preserved by Tr^r. It follows that

Tr^r: Hom_GLn Vn λH_n⁰ λ Hom_GLn 1 Vn 1 λH_n⁰ 1 λ

is nonzero. Since the last space is one dimensional, we are done.

THEOREM8.11. (Donkin 1993) Suppose that M has a Weyl filtration and that N has a good filtration. Then

Tr^r: HomGLn MN HomGLn 1 Tr^rMTr^rN is surjective.

PROOF. The proof runs by induction in the number of subquotients in the filtrations of M,N. Suppose first thatM Vn λ for some dominantλ. The result follows from Lemma 8.10 in caseNis an induced module. So we may assume the existence ofN,N with good filtrations, so that

0 N N N 0

74 8. SCHUR-WEYL DUALITY

is exact. By exactness of Tr^rwe get

0 Tr^rN Tr^rN Tr^rN 0

By Corollary 2.3 both Ext¹_GL

n

Vn

λN 0 and Ext¹_GL

n 1

Tr^rVn

λTr^rN 0. Ap-plying HomGLn

Vn

λ and HomGLn 1

Tr^rVn

λ to the exact sequences above (and omitting the subscripts on Hom) we get

0 HomVn

λN HomVn

λN HomVn

λN 0

0 HomTr^rVn

λTr^rN HomTr^rVn

λTr^rN HomTr^rVn

λTr^rN 0 The first and last vertical map are surjective by induction. It follows by the snake lemma that the middle map is surjective.

We leave the general case to the reader.

PROPOSITION8.12. Letλ Pn have degree r.

(i) Ifλ Pn 1 thenTr^rTn

λ Tn 1

λ, (ii) Ifλ Pn 1 thenTr^rTn

λ 0.

PROOF. Let us consider (ii) first. A Weyl factorVn

µ inTn

λ has highest weight µ λ, soµ Pn 1 . From Proposition 8.8 we see that all Weyl factors inTn

λ get killed by Tr^r.

Proposition 8.7 shows that Tr^rTn

λ is tilting. It is clear that this module has highest weightλ ifλ Pn 1. So we must show that it is indecomposable. It follows from Theorem 8.11 that

End_GLn

Tn

λ End_GLn 1

Tr^rTn

λ

is surjective. It is also a ring homomorphism: Recall that the functor Tr^r is composed of truncation to degreer, restriction to GLn 1, and truncation to degreer. Each of these induces ring homomorphisms; for a GLn-moduleM, End_GLn

M End_GLn

M^r is a ring homomorphism because there is no homomorphisms between modules in unequal degree. It remains to note that the surjective image of a local ring is local.

Schur-Weyl duality, part one

We now return to the ring homomorphism (8.1) considered in the beginning of this chapter. We shall prove the following theorem in the course of this section.

THEOREM8.13. The ring homomorphism kΣr End_GLn

N ^r is surjective for all values of r and n.

We begin with a proposition, that proves one half of the theorem.

PROPOSITION8.14. (Carter and Lusztig 1974) Assume r n. Then kΣr^"! End_GLn

N ^r

PROOF. We fix a basis^# e1en^$ ofN, so thateihas weightεi. The corresponding basis ofN ^r is denoted byei ei₁%'&&&% eir. Fort ⁽ 0 we lethl

t denote the diagonal matrix withεj

hl

t 1 for j ⁽ l, andεl

hl

t t. Thenhl

te_i t^)i:l*e_iwherei:l

#^# ij⁺ij l^$ . Thereforehl

t determines the number of times a specificelappears in a basis vectore_i. This is needed in the proof.

We prove surjectivity first. Choose φ End_GLn

N ^r , leti 1n, and write φei ∑jcjiej. Using GLn-linearity it follows that

∑

j

c_jit^)i:l*e_j φhl

te_i hl

tφe_i

∑

j

c_jihl

te_j

∑

j

c_jit^)j:l*e_j

SCHUR-WEYL DUALITY, PART TWO 75

Comparing the coefficients of ej, we see thatcjit^i:l cjit^j:l. This holds for anyt 0 and anyl, and we conclude thatcji 0 implies i:l j:l for eachl. This shows that

j1 jr is a permutation of 1 n, which is denotedσ. We now have φei

∑

σ Σr

cσiiσe_i (8.5)

It remains to prove thatφe_j ∑σ Σ_rcσiiσe_jfor all basis vectorse_jofN ^r.

Choose a basis vector e_j of N ^r. We define an endomorphism, xon N byxe1

ej₁ xer ej_rxer 1 0 (recall that r n). Thenx becomes an endomorphism on N ^r, andxe_i e_j. Recall that the actions of GLn andΣr on N ^r commutes; as endomorphisms of Nwe have kGLn EndN so xcommutes with any σ kΣr as endomorphisms ofN ^r. Then

φej φxei x

∑

σ Σn

cσiiσei

∑

σ Σn

cσiiσ ej

This proves surjectivity. Injectivity follows at once: Assume 0 ∑_σ Σncσσ. Then in particular

0

∑

σ Σ_ncσσe1 en

But σe1 en are non-equal basis vectors ofN ^r, and therefore eachcσ 0.

LetMdenote an vector space of dimensionn 1 overkwith automorphism group GLn 1. We embed GLn 1 in GLn as in the previous section. We apply the results on restrictions to GLn 1 to obtain the second half of the proof of Theorem 8.13.

PROPOSITION8.15. We have a surjective ring homomorphism EndGLn

N ^r EndGLn 1

M ^r

PROOF. Considered as a GLn 1-module we haveN M^! kas they are tilting modules with equal characters. It follows that the GLn 1-submodule in degreer of N ^r M^! k ^risM ^r. Recall the functor Tr^r. We see that Tr^rN ^r M ^r, so that Tr^r provide us with the homomorphism in the proposition. Surjectivity follows from Theorem 8.11. It is a ring homomorphism by the proof of Proposition 8.12.

PROOF OFTHEOREM8.13. This proof is merely a restatement of Propositions 8.14 and 8.15. Ifr nthe theorem follows immediately from the first, and ifr^" nthen applying the second a number of times gives a surjective homomorphism:

kΣr^# End_GLr

R ^r End_GLn

N ^r$

HereRis the naturalr-dimensional GLr-module.

Schur-Weyl duality, part two

Recall that the partitionn1^%&'% nnis identified with the weightn1ε1⁽⁽ nnεn. THEOREM8.16. Letλdenote a p-regular partition of r. Then

dimD^λ N ^r:Tn

λ .

We prepare the proof of Theorem 8.16 with the important Proposition 8.17. The surjectionkΣr End_GLn

N ^r of Theorem 8.13 allow us to consider End_GLn

N ^r -modules as representations of the symmetric groupΣr. For eachλ X the spaceN ^rU_λ⁾ (the U -fixpoints of weightλ) is preserved by all GLn-linear maps; it is therefore a End_GLn

N ^r-module. In general, it is not simple but we will produce a simple quotient of this module. This is done in Proposition 8.17 below, that describes the structure of the End_GLn

N ^r-modulesN ^rU)_λ .

76 8. SCHUR-WEYL DUALITY

The results in Proposition 8.17 below holds, however, in the more general context of EndGLn

Q-modules, whereQis an arbitrary tilting module. And we will, in fact, need the results of Proposition 8.17 again in the next chapter, Chapter 9, recasted for anotherQ.

Therefore the following proposition is formulated in terms of EndGLn

Q-modules with Qan arbitrary tilting module. Also, we note that the proof of Proposition 8.17 below does not depend on the group GLn; but the applications of it, in this and the following chapter, is limited to GLn. For the purpose of this chapter the reader is invited to mentally replace eachQbyN ^r.

We follow the approach of (Mathieu 2000); see in particular Lemma 11.1 inloc.cit.

Leta_λdenote the multiplicity Q:Tn

λ. For each dominantλ, definei_λ andp_λas the inclusion and projection ofa_λTn

λ inQ.

Q µaµTn

µ ^pλ ////

aλTn

λ

?_

i_λ

oo

In the proposition below we abuse notation and write p_λalso for the restriction of p_λ to Q^U_λ .

P^ROPOSITION8.17.

(i) pλ : Q^U_λ

aλTn

λ^U_λ is a surjective homomorphism of EndGLn

Q -modules.

(ii) Each non-zero element of aλTn

λ_λgenerates Q^U_λ asEndGLn

Q-module.

(iii) a_λTn

λ^U_λ is a simpleEnd_GLn

Q-module of dimension a_λ.

PROOF. Clearly the map in (i) is surjective . To show the first assertion it is enough to see that the restriction ofp_λis End_GLn

Q -linear. The endomorphismσ End_GLn

Q acts by restriction on a_λTn

λ^U_λ , that is as p_λσi_λ. To prove the proposition we must show that pλσv pλσi_λpλ

v for eachv Q^U_λ . Recall that HomVn

λQ is isomorphic toQ^U_λ and that the isomorphism is given by evaluation in a fixed nonzero element ofVn

λ_λ. Any map in HomVn

λQ lifts to HomTn

λQ, sinceVn

λ is a submodule ofTn

λ, the quotientTn

λ Vn

λ has a Weyl filtration, andQhas a good filtration. It follows that eachv Q^U_λ is in the image of some F HomTn

λQ.

This leads us to consider the mappλσF. Corresponding to the decomposition ofQwe have a decomposition of the identity, IdQ ∑µiµpµ. ThuspλσF ∑µpλσiµpµF. Consider one term in the sum

Tn

λ ^pµF //aµTn

µ ^pλσiµ //aλTn

λ (8.6)

If pλσv 0 then the map (8.6) is nonzero on the λ-weight space ofTn

λ and have nonzero image in one of the summands ofaλTn

λ . The result is a mapTn

λ

Tn

λ which is nonzero on theλ-weight space, hence not nilpotent. An endomorphism of an in-decomposable module is either nilpotent or an automorphism (Fittings lemma). It follows thatTn

λ is a summand ofaµTn

µ, hence thatλ µ. We have shown that pλσF pλσiλpλF

which implies End_GLn

Q-linearity of the restriction ofp_λtoQ^U_λ . We have proved (i).

We prove (ii). Choose a non-zero elementtλ Tλλ. SinceTλ is a summand ofQ we may considertλas an element ofQ. We claim thattλgeneratesQ^U_λ as a EndGLn

Q -module. To prepare the argument, choose a generatorv_λ Vλ_λof the Weyl module with highest weightλ. SinceVλ is a submodule ofTλ we may fix an embedding that maps v_λtot_λ.

So letx Q^U_λ be arbitrary. We construct aσ EndGLn

Q that mapstλtox. This will show thatt_λgeneratesQ^U_λ as a End_GLn

Q-module.

In document Afhandling (Sider 69-81)