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Models and refined models for involutory reflection groups and classical Weyl groups Models and refined models for involutory reflection groups and classical Weyl groups FABRIZIO CASELLI AND ROBERTA FULCI FPSAC10 - San Francisco Models and
Models and refined models for involutory reflection groups and classical Weyl groups
FABRIZIO CASELLI AND ROBERTA FULCI FPSAC’10 - San Francisco
Models and refined models for involutory reflection groups and classical Weyl groups
classical complex reflection groups;
Models and refined models for involutory reflection groups and classical Weyl groups
classical complex reflection groups;
bigger than {G(r, n)}. Build a Gelfand model for these groups;
Models and refined models for involutory reflection groups and classical Weyl groups
classical complex reflection groups;
bigger than {G(r, n)}. Build a Gelfand model for these groups;
reflection groups. Two examples: 3a Bn; 3b Dn.
Models and refined models for involutory reflection groups and classical Weyl groups
→ 1. Introduce projective reflection groups, a generalization of classical complex reflection groups
Models and refined models for involutory reflection groups and classical Weyl groups
Let V be a ℂ-vector space of finite dimension.
Models and refined models for involutory reflection groups and classical Weyl groups
Let V be a ℂ-vector space of finite dimension. A complex reflection group is a group G < GL(V ) which is generated by reflections.
Models and refined models for involutory reflection groups and classical Weyl groups
Let V be a ℂ-vector space of finite dimension. A complex reflection group is a group G < GL(V ) which is generated by reflections. Finite complex reflection groups consist of (Shephard-Todd, 1954; Chevalley, 1955): an infinite family of groups G(r, p, n), where p∣r; 34 more sporadic groups.
Models and refined models for involutory reflection groups and classical Weyl groups
Let V be a ℂ-vector space of finite dimension. A complex reflection group is a group G < GL(V ) which is generated by reflections. Finite complex reflection groups consist of (Shephard-Todd, 1954; Chevalley, 1955): an infinite family of groups G(r, p, n), where p∣r; 34 more sporadic groups. Let us focus on G(r, p, n).
Models and refined models for involutory reflection groups and classical Weyl groups
G(r, p, n) is the group of monomial matrices such that: the matrix is n × n; its non-zero entries are rth roots of 1; its permanent is a r
p th root of 1.
Models and refined models for involutory reflection groups and classical Weyl groups
G(r, p, n) is the group of monomial matrices such that: the matrix is n × n; its non-zero entries are rth roots of 1; its permanent is a r
p th root of 1.
Example g = ⎛ ⎜ ⎜ ⎝ 1 −1 1 1 ⎞ ⎟ ⎟ ⎠ ∈ G(2, 1, 4) = B4 / ∈ G(2, 2, 4) = D4
Models and refined models for involutory reflection groups and classical Weyl groups
G(r, p, n) is the group of monomial matrices such that: the matrix is n × n; its non-zero entries are rth roots of 1; its permanent is a r
p th root of 1.
Example g = ⎛ ⎜ ⎜ ⎝ 1 −1 1 1 ⎞ ⎟ ⎟ ⎠ ∈ G(2, 1, 4) = B4 / ∈ G(2, 2, 4) = D4 h = ⎛ ⎜ ⎜ ⎝ 1 −1 −1 1 ⎞ ⎟ ⎟ ⎠ ∈ G(2, 2, 4) = D4
Models and refined models for involutory reflection groups and classical Weyl groups
Let W be a finite complex reflection group.
Models and refined models for involutory reflection groups and classical Weyl groups
Let W be a finite complex reflection group. If W > Cq = ⟨e
2휋i q Id⟩, we can consider its quotient W
Cq acting on the symmetric power Sq(V ) of V .
Models and refined models for involutory reflection groups and classical Weyl groups
Let W be a finite complex reflection group. If W > Cq = ⟨e
2휋i q Id⟩, we can consider its quotient W
Cq acting on the symmetric power Sq(V ) of V . Definition (Caselli, 2009) G < GL(Sq(V )) is a projective reflection group if there exists a reflection group W such that G = W Cq .
Models and refined models for involutory reflection groups and classical Weyl groups
Let W be a finite complex reflection group. If W > Cq = ⟨e
2휋i q Id⟩, we can consider its quotient W
Cq acting on the symmetric power Sq(V ) of V . Definition (Caselli, 2009) G < GL(Sq(V )) is a projective reflection group if there exists a reflection group W such that G = W Cq . We will focus on the case W = G(r, p, n).
Models and refined models for involutory reflection groups and classical Weyl groups
G(r, p, q, n) def = G(r, p, n) Cq
Models and refined models for involutory reflection groups and classical Weyl groups
G(r, p, q, n) def = G(r, p, n) Cq G(r, p, q, n) is well defined if:
Models and refined models for involutory reflection groups and classical Weyl groups
G(r, p, q, n) def = G(r, p, n) Cq G(r, p, q, n) is well defined if: p∣r q∣r pq∣rn
Models and refined models for involutory reflection groups and classical Weyl groups
G(r, p, q, n) def = G(r, p, n) Cq G(r, p, q, n) is well defined if: p∣r q∣r pq∣rn These conditions are symmetric in p and q, so...
Models and refined models for involutory reflection groups and classical Weyl groups
G(r, p, q, n) def = G(r, p, n) Cq G(r, p, q, n) is well defined if: p∣r q∣r pq∣rn These conditions are symmetric in p and q, so... G(r, p, q, n) exists ⇔ G(r, q, p, n) exists.
Models and refined models for involutory reflection groups and classical Weyl groups
Definition Let G = G(r, p, q, n). Its dual group G ∗ is G(r, q, p, n).
Models and refined models for involutory reflection groups and classical Weyl groups
Definition Let G = G(r, p, q, n). Its dual group G ∗ is G(r, q, p, n). Many algebraic objects related to the group G(r, p, q, n) find a natural description via the combinatorics of G ∗.
Models and refined models for involutory reflection groups and classical Weyl groups
Definition Let G = G(r, p, q, n). Its dual group G ∗ is G(r, q, p, n). Many algebraic objects related to the group G(r, p, q, n) find a natural description via the combinatorics of G ∗. For example, its representation theory.
Models and refined models for involutory reflection groups and classical Weyl groups
→ 2. introduce involutory reflection groups. Build a Gelfand model for these groups
Models and refined models for involutory reflection groups and classical Weyl groups
A Gelfand model of a group G is a G-module containing each irreducible complex representation of G exactly once:
Models and refined models for involutory reflection groups and classical Weyl groups
A Gelfand model of a group G is a G-module containing each irreducible complex representation of G exactly once: (M, 휌) ∼ = ⊕
휙∈Irr(G)
(V휙, 휙) Irr(G) = {irreducible representations of G}.
Models and refined models for involutory reflection groups and classical Weyl groups
Inglis-Richardson-Saxl, for symmetric groups; Kodiyalam-Verma, for symmetric groups; Aguado-Araujo-Bigeon, for Weyl groups; Baddeley, for wreath products; Adin-Postnikov-Roichman, for the groups G(r, n)...
Models and refined models for involutory reflection groups and classical Weyl groups
Inglis-Richardson-Saxl, for symmetric groups; Kodiyalam-Verma, for symmetric groups; Aguado-Araujo-Bigeon, for Weyl groups; Baddeley, for wreath products; Adin-Postnikov-Roichman, for the groups G(r, n)... Caselli, for all involutory reflection groups.
Models and refined models for involutory reflection groups and classical Weyl groups
Definition Let g ∈ G < GL(n, ℂ). g is an absolute involution if g ¯ g = 1, ¯ g being the complex conjugate of g.
Models and refined models for involutory reflection groups and classical Weyl groups
Definition Let g ∈ G < GL(n, ℂ). g is an absolute involution if g ¯ g = 1, ¯ g being the complex conjugate of g. Definition Let G < GL(n, ℂ) and let M be a Gelfand model for G. G is involutory if dim(M) = #{ absolute involutions of G}.
Models and refined models for involutory reflection groups and classical Weyl groups
Theorem (Caselli, 2009) A group G(r, p, n) is involutory if and only if GCD(p, n) = 1, 2.
Models and refined models for involutory reflection groups and classical Weyl groups
Theorem (Caselli, 2009) A group G(r, p, n) is involutory if and only if GCD(p, n) = 1, 2. I(r, p, q, n) = {absolute involutions of G(r, p, q, n)}
Models and refined models for involutory reflection groups and classical Weyl groups
Theorem (Caselli, 2009) A group G(r, p, n) is involutory if and only if GCD(p, n) = 1, 2. I(r, p, q, n) = {absolute involutions of G(r, p, q, n)} #I(r, p, q, n) = #I(r, q, p, n)
Models and refined models for involutory reflection groups and classical Weyl groups
Caselli’s model (M, 휌) for an involutory group G = G(r, p, q, n) looks like this:
Models and refined models for involutory reflection groups and classical Weyl groups
Caselli’s model (M, 휌) for an involutory group G = G(r, p, q, n) looks like this: M is the vector space spanned by the absolute involutions of G ∗: M = ⊕
v∈I(r,q,p,n)
ℂ Cv
Models and refined models for involutory reflection groups and classical Weyl groups
Caselli’s model (M, 휌) for an involutory group G = G(r, p, q, n) looks like this: M is the vector space spanned by the absolute involutions of G ∗: M = ⊕
v∈I(r,q,p,n)
ℂ Cv the morphism 휌 : G → GL(M) has the form 휌(g)v = 휙v(g)C∣g∣v∣g∣−1, 휙v(g) being a scalar.
Models and refined models for involutory reflection groups and classical Weyl groups
reflection groups. Two examples: → 3a Bn; 3b Dn.
Models and refined models for involutory reflection groups and classical Weyl groups
g = (2, −4, 3, 1) ∈ G(2, 1, 4) = B4. split g into two double-rowed vectors according to the sign: g0 = ( 1 3 4 2 3 1 ) g1 = ( 2 4 )
Models and refined models for involutory reflection groups and classical Weyl groups
g = (2, −4, 3, 1) ∈ G(2, 1, 4) = B4. split g into two double-rowed vectors according to the sign: g0 = ( 1 3 4 2 3 1 ) g1 = ( 2 4 ) perform RS to the two double-rowed vectors: g0
RS
− → (P0, Q0) = ( 1 3 2 , 1 3 4 ) ; g1
RS
− → (P1, Q1) = ( 4 , 2 )
Models and refined models for involutory reflection groups and classical Weyl groups
g = (2, −4, 3, 1) ∈ G(2, 1, 4) = B4. split g into two double-rowed vectors according to the sign: g0 = ( 1 3 4 2 3 1 ) g1 = ( 2 4 ) perform RS to the two double-rowed vectors: g0
RS
− → (P0, Q0) = ( 1 3 2 , 1 3 4 ) ; g1
RS
− → (P1, Q1) = ( 4 , 2 ) glue the images of g0 and g1 together: g
RS
− → (P0, P1; Q0, Q1) = ( 1 3 2 , 4 ; 1 3 4 , 2 )
Models and refined models for involutory reflection groups and classical Weyl groups
g ∈ B∗
n = Bn is an (absolute) involution if and only if
g
RS
− → (P0, P1; P0, P1).
Models and refined models for involutory reflection groups and classical Weyl groups
g ∈ B∗
n = Bn is an (absolute) involution if and only if
g
RS
− → (P0, P1; P0, P1). Thus our model for Bn is spanned by the elements {g ∈ Bn : g
RS
− → (P0, P1; P0, P1)} =
Models and refined models for involutory reflection groups and classical Weyl groups
g ∈ B∗
n = Bn is an (absolute) involution if and only if
g
RS
− → (P0, P1; P0, P1). Thus our model for Bn is spanned by the elements {g ∈ Bn : g
RS
− → (P0, P1; P0, P1)} = = { symmetric matrices of Bn} =: Sym(Bn).
Models and refined models for involutory reflection groups and classical Weyl groups
There is a nice parametrization for Bn’s representations: {irreducible representations of Bn}
Models and refined models for involutory reflection groups and classical Weyl groups
There is a nice parametrization for Bn’s representations: {irreducible representations of Bn}
{ordered pairs of Ferrers diagrams (휆, 휇) such that ∣휆∣ + ∣휇∣ = n}
Models and refined models for involutory reflection groups and classical Weyl groups
The irreducible representations of B3 are: , ∅ , ∅ , ∅ , , ∅, ∅, ∅, , ,
Models and refined models for involutory reflection groups and classical Weyl groups
Recall that the representation that makes M a model has the form 휌(g)v = 휙v(g)C∣g∣v∣g∣−1.
Models and refined models for involutory reflection groups and classical Weyl groups
Recall that the representation that makes M a model has the form 휌(g)v = 휙v(g)C∣g∣v∣g∣−1. Definition Two elements of Bn are Sn-conjugate if they are conjugate via an element of Sn.
Models and refined models for involutory reflection groups and classical Weyl groups
Recall that the representation that makes M a model has the form 휌(g)v = 휙v(g)C∣g∣v∣g∣−1. Definition Two elements of Bn are Sn-conjugate if they are conjugate via an element of Sn. Thus M naturally splits into submodules M(c), where each c is a Sn-conjugacy class of involutions of Bn.
Models and refined models for involutory reflection groups and classical Weyl groups
Which of the irreducible representations of Bn are afforded by each
Models and refined models for involutory reflection groups and classical Weyl groups
Which of the irreducible representations of Bn are afforded by each
It is quite natural to expect the decomposition to be well behaved with respect to the RS correspondence.
Models and refined models for involutory reflection groups and classical Weyl groups
Which of the irreducible representations of Bn are afforded by each
It is quite natural to expect the decomposition to be well behaved with respect to the RS correspondence. And so it is!
Models and refined models for involutory reflection groups and classical Weyl groups
v
RS
− → (P0, P1; P0, P1)
Models and refined models for involutory reflection groups and classical Weyl groups
v
RS
− → (P0, P1; P0, P1) Sh(v) = shape of (P0, P1)
Models and refined models for involutory reflection groups and classical Weyl groups
v
RS
− → (P0, P1; P0, P1) Sh(v) = shape of (P0, P1) Theorem (Caselli, F., 2010) Let c be a Sn-conjugacy class of involutions in Bn.
Models and refined models for involutory reflection groups and classical Weyl groups
v
RS
− → (P0, P1; P0, P1) Sh(v) = shape of (P0, P1) Theorem (Caselli, F., 2010) Let c be a Sn-conjugacy class of involutions in Bn. The following decomposition holds: M(c) ∼ = ⊕
(휆,휇)∈Sh(c)
휌휆,휇, where
Models and refined models for involutory reflection groups and classical Weyl groups
v
RS
− → (P0, P1; P0, P1) Sh(v) = shape of (P0, P1) Theorem (Caselli, F., 2010) Let c be a Sn-conjugacy class of involutions in Bn. The following decomposition holds: M(c) ∼ = ⊕
(휆,휇)∈Sh(c)
휌휆,휇, where Sh(c) = ∪
v∈c
Sh(v).
Models and refined models for involutory reflection groups and classical Weyl groups
In words:
Models and refined models for involutory reflection groups and classical Weyl groups
In words: if a submodule M(c) of M is spanned by involutions whose images via RS have certain shapes...
Models and refined models for involutory reflection groups and classical Weyl groups
In words: if a submodule M(c) of M is spanned by involutions whose images via RS have certain shapes... ...M(c) affords the irreducible representations of Bn parametrized by those shapes.
Models and refined models for involutory reflection groups and classical Weyl groups
Two involutions v and w of Bn are Sn-conjugate if and only if v
RS
− → (P0, P1; P0, P1) w
RS
− → (Q0, Q1; Q0, Q1) with:
Models and refined models for involutory reflection groups and classical Weyl groups
Two involutions v and w of Bn are Sn-conjugate if and only if v
RS
− → (P0, P1; P0, P1) w
RS
− → (Q0, Q1; Q0, Q1) with: P0 and Q0 have the same number of boxes; P1 and Q1 have the same number of boxes; P0 and Q0 have the same number of columns of odd length; P1 and Q1 have the same number of columns of odd length.
Models and refined models for involutory reflection groups and classical Weyl groups
v = (−6, 4, 3, 2, −5, −1, −7) ∈ B7. Let c be the S7-conjugacy class of v. Then
Models and refined models for involutory reflection groups and classical Weyl groups
v = (−6, 4, 3, 2, −5, −1, −7) ∈ B7. Let c be the S7-conjugacy class of v. Then M(c) ∼ = ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) .
Models and refined models for involutory reflection groups and classical Weyl groups
reflection groups. Two examples: 3a Bn; →3b Dn.
Models and refined models for involutory reflection groups and classical Weyl groups
n = Bn ±Id
g ∈ Bn
RS
− → (P0, P1; Q0, Q1)
Models and refined models for involutory reflection groups and classical Weyl groups
n = Bn ±Id
g ∈ Bn
RS
− → (P0, P1; Q0, Q1) −g ∈ Bn
RS
− → (P1, P0; Q1, Q0)
Models and refined models for involutory reflection groups and classical Weyl groups
n = Bn ±Id
g ∈ Bn
RS
− → (P0, P1; Q0, Q1) −g ∈ Bn
RS
− → (P1, P0; Q1, Q0) ˜ g ∈ Bn ±Id
RS2
− → ({P0, P1}; {Q0, Q1}) ↑ ↑ UNORDERED PAIRS!
Models and refined models for involutory reflection groups and classical Weyl groups
g ∈ D∗
n is an absolute involution if and only if
g
RS2
− → {P0, P1; P0, P1}.
Models and refined models for involutory reflection groups and classical Weyl groups
g ∈ D∗
n is an absolute involution if and only if
g
RS2
− → {P0, P1; P0, P1}. Thus, our model for Dn is spanned by the elements { g ∈ D∗
n : g RS2
− → {P0, P1; P0, P1} } .
Models and refined models for involutory reflection groups and classical Weyl groups
Model for Dn spanned by {˜ g ∈ Bn ±Id : ˜ g
RS2
− → ({P0, P1}; {P0, P1})} = ↑ ↑ the same unordered pair
Models and refined models for involutory reflection groups and classical Weyl groups
Model for Dn spanned by {˜ g ∈ Bn ±Id : ˜ g
RS2
− → ({P0, P1}; {P0, P1})} = ↑ ↑ the same unordered pair = ⎧ ⎨ ⎩
Models and refined models for involutory reflection groups and classical Weyl groups
Model for Dn spanned by {˜ g ∈ Bn ±Id : ˜ g
RS2
− → ({P0, P1}; {P0, P1})} = ↑ ↑ the same unordered pair = ⎧ ⎨ ⎩ Sym ( Bn ±Id ) = {˜ g ∈
Bn ±Id : g RS
− → (P0, P1; P0, P1) for g lift of ˜ g}
Models and refined models for involutory reflection groups and classical Weyl groups
Model for Dn spanned by {˜ g ∈ Bn ±Id : ˜ g
RS2
− → ({P0, P1}; {P0, P1})} = ↑ ↑ the same unordered pair = ⎧ ⎨ ⎩ Sym ( Bn ±Id ) = {˜ g ∈
Bn ±Id : g RS
− → (P0, P1; P0, P1) for g lift of ˜ g} Asym ( Bn ±Id ) = {˜ g ∈
Bn ±Id : g RS
− → (P0, P1; P1, P0) for g lift of ˜ g}.
Models and refined models for involutory reflection groups and classical Weyl groups
v → ( 1 3 , 2 4 ; 2 4 , 1 3 )
Models and refined models for involutory reflection groups and classical Weyl groups
v → ( 1 3 , 2 4 ; 2 4 , 1 3 ) v = (−2, 1, −4, 3) = ⎛ ⎜ ⎜ ⎝ −1 1 −1 1 ⎞ ⎟ ⎟ ⎠
Models and refined models for involutory reflection groups and classical Weyl groups
v → ( 1 3 , 2 4 ; 2 4 , 1 3 ) v = (−2, 1, −4, 3) = ⎛ ⎜ ⎜ ⎝ −1 1 −1 1 ⎞ ⎟ ⎟ ⎠ ˜ v ∈ Asym ( B4 ±Id )
Models and refined models for involutory reflection groups and classical Weyl groups
Irreducible representations of Dn:
Models and refined models for involutory reflection groups and classical Weyl groups
Irreducible representations of Dn: {휆, 휇}, with 휆 ∕= 휇, ∣휆∣ + ∣휇∣ = n (UNSPLIT REP);
Models and refined models for involutory reflection groups and classical Weyl groups
Irreducible representations of Dn: {휆, 휇}, with 휆 ∕= 휇, ∣휆∣ + ∣휇∣ = n (UNSPLIT REP); {휆, 휆}+, with 휆 ⊢ n
2 (SPLIT REP);
{휆, 휆}−, with 휆 ⊢ n
2 (SPLIT REP)
Models and refined models for involutory reflection groups and classical Weyl groups
Irreducible representations of Dn: {휆, 휇}, with 휆 ∕= 휇, ∣휆∣ + ∣휇∣ = n (UNSPLIT REP); {휆, 휆}+, with 휆 ⊢ n
2 (SPLIT REP);
{휆, 휆}−, with 휆 ⊢ n
2 (SPLIT REP)
Possible shapes via RS2 of the generators of M: {휆, 휇}, with 휆 ∕= 휇, ∣휆∣ + ∣휇∣ = n (SYMMETRIC GEN);
Models and refined models for involutory reflection groups and classical Weyl groups
Irreducible representations of Dn: {휆, 휇}, with 휆 ∕= 휇, ∣휆∣ + ∣휇∣ = n (UNSPLIT REP); {휆, 휆}+, with 휆 ⊢ n
2 (SPLIT REP);
{휆, 휆}−, with 휆 ⊢ n
2 (SPLIT REP)
Possible shapes via RS2 of the generators of M: {휆, 휇}, with 휆 ∕= 휇, ∣휆∣ + ∣휇∣ = n (SYMMETRIC GEN); {휆, 휆}, with 휆 ⊢ n
2 (SYMMETRIC GEN);
{휆, 휆}, with 휆 ⊢ n
2 (ANTISYMMETRIC GEN).
Models and refined models for involutory reflection groups and classical Weyl groups
Irreducible representations of Dn: {휆, 휇}, with 휆 ∕= 휇, ∣휆∣ + ∣휇∣ = n (UNSPLIT REP); {휆, 휆}+, with 휆 ⊢ n
2 (SPLIT REP);
{휆, 휆}−, with 휆 ⊢ n
2 (SPLIT REP)
Possible shapes via RS2 of the generators of M: {휆, 휇}, with 휆 ∕= 휇, ∣휆∣ + ∣휇∣ = n (SYMMETRIC GEN); {휆, 휆}, with 휆 ⊢ n
2 (SYMMETRIC GEN);
{휆, 휆}, with 휆 ⊢ n
2 (ANTISYMMETRIC GEN).
∙ In fact, a pair (P0, P1; P1, P0) can be the RS image of a g ∈ Bn
Models and refined models for involutory reflection groups and classical Weyl groups
M naturally splits first of all into the two fat submodules ⊕
v∈Sym
ℂ Cv ⊕
v∈Asym
ℂ Cv
Models and refined models for involutory reflection groups and classical Weyl groups
Again, this decomposition is well-behaved w.r.t. the RS2 correspondence!
Models and refined models for involutory reflection groups and classical Weyl groups
Again, this decomposition is well-behaved w.r.t. the RS2 correspondence! Theorem (Caselli, F., 2010) The split representations of Dn can be labelled in such a way that ⊕
v∈Asym
ℂ Cv ≃ ⊕
휆⊢ n
2
{휆, 휆}−,
Models and refined models for involutory reflection groups and classical Weyl groups
Again, this decomposition is well-behaved w.r.t. the RS2 correspondence! Theorem (Caselli, F., 2010) The split representations of Dn can be labelled in such a way that ⊕
v∈Asym
ℂ Cv ≃ ⊕
휆⊢ n
2
{휆, 휆}−, ⇓ ⊕
v∈Sym
ℂ Cv ≃ ⊕
휆∕=휇
{휆, 휇} ⊕ ⊕
휆⊢ n
2
{휆, 휆}+.
Models and refined models for involutory reflection groups and classical Weyl groups
v = (−6, 4, 3, 2, −5, −1) ∈ B6. Let ˜ c be the S6-conjugacy class of ˜
Models and refined models for involutory reflection groups and classical Weyl groups
v = (−6, 4, 3, 2, −5, −1) ∈ B6. Let ˜ c be the S6-conjugacy class of ˜
M(˜ c) ∼ = ( , ) ⊕ ( , )
+ ⊕
( , )
+.
Models and refined models for involutory reflection groups and classical Weyl groups