SchurWeyl duality over arbitrary commutative rings Tiago Cruz - - PowerPoint PPT Presentation

Schur–Weyl duality over arbitrary commutative rings

Tiago Cruz 11/06/2019

Spa, Belgium

Classical Schur–Weyl duality

V = Rn GLn(R) - the general linear group of degree n over R Sd - the symmetric group on d letters

2

Classical Schur–Weyl duality

V = Rn GLn(R) - the general linear group of degree n over R Sd - the symmetric group on d letters V ⊗d GLn(R) Sd GLn(R) ∋ g : g(v1 ⊗ · · · ⊗ vd) = gv1 ⊗ · · · ⊗ gvd Sd ∋ σ: (v1 ⊗ · · · ⊗ vd)σ = vσ(1) ⊗ · · · ⊗ vσ(d)

2

Classical Schur–Weyl duality

V = Rn GLn(R) - the general linear group of degree n over R Sd - the symmetric group on d letters V ⊗d GLn(R) Sd GLn(R) ∋ g : g(v1 ⊗ · · · ⊗ vd) = gv1 ⊗ · · · ⊗ gvd Sd ∋ σ: (v1 ⊗ · · · ⊗ vd)σ = vσ(1) ⊗ · · · ⊗ vσ(d) Extending the actions to the group algebras we obtain the algebra homomorphisms ρ: RGLn(R) → EndR

• V ⊗d

ψ: RSd → EndR

• V ⊗d

.

2

Classical Schur–Weyl duality

V = Rn GLn(R) - the general linear group of degree n over R Sd - the symmetric group on d letters V ⊗d GLn(R) Sd GLn(R) ∋ g : g(v1 ⊗ · · · ⊗ vd) = gv1 ⊗ · · · ⊗ gvd Sd ∋ σ: (v1 ⊗ · · · ⊗ vd)σ = vσ(1) ⊗ · · · ⊗ vσ(d) Extending the actions to the group algebras we obtain the algebra homomorphisms ρ: RGLn(R) → EndRSd

• V ⊗d

ψ: RSd → EndRGLn(R)

• V ⊗d

.

2

Classical Schur–Weyl duality

V = Rn GLn(R) - the general linear group of degree n over R Sd - the symmetric group on d letters V ⊗d GLn(R) Sd GLn(R) ∋ g : g(v1 ⊗ · · · ⊗ vd) = gv1 ⊗ · · · ⊗ gvd Sd ∋ σ: (v1 ⊗ · · · ⊗ vd)σ = vσ(1) ⊗ · · · ⊗ vσ(d) Extending the actions to the group algebras we obtain the algebra homomorphisms ρ: RGLn(R) → EndRSd

• V ⊗d

ψ: RSd → EndRGLn(R)

• V ⊗d

. The endomorphism algebra SR(n, d) = EndRSd

• V ⊗d

is called Schur algebra.

2

Classical Schur–Weyl duality

V ⊗d GLn(R) Sd GLn(R) ∋ g : g(v1 ⊗ · · · ⊗ vd) = gv1 ⊗ · · · ⊗ gvd Sd ∋ σ: (v1 ⊗ · · · ⊗ vd)σ = vσ(1) ⊗ · · · ⊗ vσ(d) Extending the actions to the group algebras we obtain the algebra homomorphisms ρ: RGLn(R) → EndRSd

• V ⊗d

ψ: RSd → EndRGLn(R)

• V ⊗d

. The endomorphism algebra SR(n, d) = EndRSd

• V ⊗d

is called Schur algebra. We say that (classical) Schur–Weyl duality holds for a ring R if the maps ρ and ψ are surjective.

2

When does Classical Schur–Weyl duality hold?

Theorem If ρ: RGLn(R) → SR(n, d) is surjective then ψ: RSd → EndRGLn(R)

• V ⊗d

is surjective.

3

When does Classical Schur–Weyl duality hold?

Theorem If ρ: RGLn(R) → SR(n, d) is surjective then ψ: RSd → EndRGLn(R)

• V ⊗d

is surjective. R∗ - set of all units of the commutative ring R;

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When does Classical Schur–Weyl duality hold?

Theorem If ρ: RGLn(R) → SR(n, d) is surjective then ψ: RSd → EndRGLn(R)

• V ⊗d

is surjective. R∗ - set of all units of the commutative ring R; Theorem ([4, Theorem 3.6.]) Let n, d ∈ N be natural numbers. Let R be a commutative ring with identity that contains a set S which has the following properties: ∀x, y ∈ S, x = y = ⇒ x − y ∈ R∗; |S| > d. Then the algebra homomorphism ρ: RGLn(R) → SR(n, d) is surjective, that is, Schur–Weyl duality holds.

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When does Classical Schur–Weyl duality hold?

Theorem If ρ: RGLn(R) → SR(n, d) is surjective then ψ: RSd → EndRGLn(R)

• V ⊗d

is surjective. R∗ - set of all units of the commutative ring R; Theorem ([4, Theorem 3.6.]) Let n, d ∈ N be natural numbers. Let R be a commutative ring with identity that contains a set S which has the following properties: ∀x, y ∈ S, x = y = ⇒ x − y ∈ R∗; |S| > d. Then the algebra homomorphism ρ: RGLn(R) → SR(n, d) is surjective, that is, Schur–Weyl duality holds. ρ: RGLn(R) → SR(n, d) is not surjective F2, n = d = 2.

3

Schur–Weyl duality between SR(n, d) and Sd

V ⊗d SR(n, d) Sd

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Schur–Weyl duality between SR(n, d) and Sd

V ⊗d SR(n, d) Sd EndRSd

• V ⊗d

= SR(n, d) ∋ η: η · (v1 ⊗ · · · ⊗ vd) = η(v1 ⊗ · · · ⊗ vd)

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Schur–Weyl duality between SR(n, d) and Sd

V ⊗d SR(n, d) Sd EndRSd

• V ⊗d

= SR(n, d) ∋ η: η · (v1 ⊗ · · · ⊗ vd) = η(v1 ⊗ · · · ⊗ vd) There is an algebra homomorphism ψ: RSd → EndSR(n,d)

• V ⊗d

.

4

Schur–Weyl duality between SR(n, d) and Sd

V ⊗d SR(n, d) Sd EndRSd

• V ⊗d

= SR(n, d) ∋ η: η · (v1 ⊗ · · · ⊗ vd) = η(v1 ⊗ · · · ⊗ vd) There is an algebra homomorphism ψ: RSd → EndSR(n,d)

• V ⊗d

. Theorem ([4, Corollary 3.5.]) ψ is surjective for every commutative ring R. If n ≥ d, then ψ is an isomorphism.

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Strength of Schur–Weyl duality

R - noetherian ring D = HomR(−, R) - standard duality

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Strength of Schur–Weyl duality

R - noetherian ring D = HomR(−, R) - standard duality The homomorphism ρ: RGLn(R) → SR(n, d) induces by restriction of scalars the functor HR : SR(n, d)-mod → (RGLn(R))d-mod

5

Strength of Schur–Weyl duality

R - noetherian ring D = HomR(−, R) - standard duality The homomorphism ρ: RGLn(R) → SR(n, d) induces by restriction of scalars the functor HR : SR(n, d)-mod → (RGLn(R))d-mod SW duality on R = ⇒ HR equivalence.

5

Strength of Schur–Weyl duality

R - noetherian ring D = HomR(−, R) - standard duality The homomorphism ρ: RGLn(R) → SR(n, d) induces by restriction of scalars the functor HR : SR(n, d)-mod → (RGLn(R))d-mod SW duality on R = ⇒ HR equivalence. ψ is related to the Schur functor F R = HomSR(n,d)(V ⊗d, −): SR(n, d)-mod → RSd-mod, n ≥ d.

5

Strength of Schur–Weyl duality

R - noetherian ring D = HomR(−, R) - standard duality HR : SR(n, d)-mod → (RGLn(R))d-mod SW duality on R = ⇒ HR equivalence. ψ is related to the Schur functor F R = HomSR(n,d)(V ⊗d, −): SR(n, d)-mod → RSd-mod, n ≥ d. ψ isomorphism = ⇒ F R

SR(n,d)-proj : SR(n, d)-proj →addRSd DV ⊗d

equivalence.

5

Strength of Schur–Weyl duality

R - noetherian ring D = HomR(−, R) - standard duality HR : SR(n, d)-mod → (RGLn(R))d-mod SW duality on R = ⇒ HR equivalence. ψ is related to the Schur functor F R = HomSR(n,d)(V ⊗d, −): SR(n, d)-mod → RSd-mod, n ≥ d. ψ isomorphism = ⇒ F R

SR(n,d)-proj : SR(n, d)-proj →addRSd DV ⊗d

equivalence. The functor G R = HomRSd(DV ⊗d, −) is right adjoint to F R. The strength

• f SW duality on SR(n, d) is measured by the degree to which G R fails to

be exact on RSd-mod.

5

Strength of Schur–Weyl duality

R - noetherian ring D = HomR(−, R) - standard duality HR : SR(n, d)-mod → (RGLn(R))d-mod SW duality on R = ⇒ HR equivalence. F R = HomSR(n,d)(V ⊗d, −): SR(n, d)-mod → RSd-mod, n ≥ d. ψ isomorphism = ⇒ F R

SR(n,d)-proj : SR(n, d)-proj →addRSd DV ⊗d

equivalence. The functor G R = HomRSd(DV ⊗d, −) is right adjoint to F R. The strength

• f SW duality on SR(n, d) is measured by the degree to which G R fails to

be exact on RSd-mod. Theorem ([6, Theorem 5.1], C.) G Z

addZSd DV ⊗d is exact. G F2 addF2Sd DV ⊗d is not exact. 5

References i

References

[1] Benson, D. and Doty, S. (2009). Schur–Weyl duality over finite

• fields. Arch. Math., 93(5):425–435.

[2] Bryant, R. M. (2009). Lie powers of infinite-dimensional modules.

• Beitr. Algebra Geom., 50(1):179–193.

[3] Carter, R. W. and Lusztig, G. (1974). On the modular representations

• f the general linear and symmetric groups. Math. Z., 136:193–242.

[4] Cruz, T. (2018). Schur–Weyl duality over commutative rings. Communications in Algebra. [5] de Concini, C. and Procesi, C. (1976). A characteristic free approach to invariant theory. Adv. Math., 21:330–354.

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References ii

[6] Fang, M. and Koenig, S. (2011). Schur functors and dominant

• dimension. Trans. Amer. Math. Soc., 363(3):1555–1576.

[7] Green, J. A. (1980). Polynomial representations of GLn, volume 830. Springer, Cham. [8] K¨

• nig, S., Slung˚

ard, I. H., and Xi, C. (2001). Double centralizer properties, dominant dimension, and tilting modules. J. Algebra, 240(1):393–412. [9] Krause, H. (2015). Polynomial representations of GL(n) and Schur-Weyl duality. Beitr. Algebra Geom., 56(2):769–773. [10] Rouquier, R. (2008). q-Schur algebras and complex reflection

• groups. Mosc. Math. J., 8(1):119–158.

[11] Schur, I. (1928). ¨ Uber die stetigen Darstellungen der allgemeinen linearen Gruppe. Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl., 1928:100–124.

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Thank You

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