PDF Invariant theory of singular α-determinants

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Invariant theory of singular α-determinants

Kazufumi KIMOTO

(University of the Ryukyus)

Joint work with Masato WAKAYAMA

Conference “Representation Theory, Systems of Diﬀerential Equations and their Related Topics” at Hokkaido University

July 5, 2007

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α-determinant

For α ∈ C and A = (a_ij)₁_≤_i,j_≤_n ∈ Mat_n = Mat_n(C), the α- determinant of A is

det^(α)(A) = X

σ∈Sn

α^ν(σ)a_σ(1)1 . . . a_σ(n)n where

ν(σ) = X

i≥1

(i −1)m_i if the cycle-type of σ is 1^m¹2^m² . . ..

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2

• det⁽⁻¹⁾(A) = det(A) (∵ (−1)^ν^(σ) = sgnσ)

• det⁽¹⁾(A) = per(A)

• det^(α) is multilinear in rows and columns

• det^(α)(^tA) = det^(α)(A) (∵ ν(σ⁻¹) = ν(σ))

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• α-determinant is ﬁrst introduced by Vere-Jones (1988) as coeﬃ- cients in the expansion of det(I − αA)⁻^1/α:

det(I − αA)⁻^1/α = X∞ n=0

1 n!

X

1≤i1,...,in≤d

det^(α)





a_i₁_i₁ . . . a_i₁_i_n ... . .. ... a_i_n_i₁ . . . a_i_n_i_n





for A = (a_ij)₁_≤_i,j_≤_d.

This is used to construct a certain point process.

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4

• det is multiplicative : det(AB) = det(A) det(B)

• det^(α)(AB) 6= det^(α)(A) det^(α)(B) if α 6= −1

det is multiplicative ←→ GL_n(C) · det(X) ⊂ C · det(X)

−→ Look at the smallest GL_n(C)-invariant subspace containing det^(α)(X)

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polynomial functions on Mat_n by E_ij =

Xn p=1

x_ip ∂

∂x_jp E_ij : standard basis of gl_n = gl_n(C)

x_ij : standard coordinate on Mat_n = Mat_n(C)

• U(gl_n) ·det(X) = C · det(X) ∼= M⁽¹ⁿ ⁿ⁾

• U(gl_n)· per(X) ∼= M⁽ⁿ⁾ⁿ

M^λn : irreducible highest weight module of U(gl_n) with highest weight λ (we identify the highest weights and partitions and/or Young diagrams)

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6

¨ ¥

§ ¦

U(gl_n) · det^(α)(X) ∼= M

λ`n f_λ(α)6=0

¡M^λn

¢⊕f^λ

where

f_λ(α) = Y

(i,j)∈λ

(1 + (j − i)α),

f^λ = K_λ,(1ⁿ₎ = # of standard tableaux of λ,

(Matsumoto-Wakayama (2006))

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λ`n λ1≤k

¡M^λn

¢⊕f^λ

• If α = 1/k (k = 1,2, . . . , n − 1),

λ`n λ⁰₁≤k

¡M^λn

¢⊕f^λ

• Otherwise,

U(gl_n) · det^(α)(X) ∼= (Cⁿ)^⊗ⁿ ∼= M

λ`n

¡M^λn

¢⊕f^λ

α = ±1,±1

2, . . . ,± 1

n − 1 : singular values

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8

Singular α-determinants

Look at the case where α = −1/k (k = 1,2, . . . , n − 1).

det^(α)





1 . . . 1 ... . .. ... 1 . . . 1



 = X

σ∈Sn

α^ν^(σ) = Y

1≤i<n

(1 +iα)

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For I ⊂ [n] = {1,2, . . . , n},

S_n(I) = {σ ∈ S_n; σ(x) = x, x /∈ I}.

σ · (a_ij) = (a_σ−1(i)j), (a_ij) · τ = (a_iτ_(j))

((a_ij) ∈ Mat_m,n, σ ∈ S_m,τ ∈ S_n) X

σ∈Sn(I)

det^(α)(X · σ)

= Y

1≤i<k

(1 +iα) X

g∈Sn

α^m(g,I)x_g(1)1. . . x_g(n)n

(m(g, I): some nonnegative integer)

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“−1/k-analogue” of the alternating property: 10

¨ ¥

§ ¦

I ⊂ [n], #I > k =⇒ X

σ∈Sn(I)

det⁽⁻^1/k)(X · σ) = 0.

In particular,

¨ ¥

§ ¦

(a₁, . . . ,a_n) ∈ Mat_n,

a_i₁ = · · · = a_i_k = b (1 ≤ i₁ < · · · < i_k ≤ n), j 6= i₁, . . . , i_k

=⇒ det⁽⁻^1/k)(a₁, . . . ,a_j + b, . . . ,a_n)

= det⁽⁻^1/k)(a₁, . . . ,a_j, . . . ,a_n)

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• The ordinary determinant is characterized (up to constant) by the multilinearity and alternating property with respect to column vectors.

• How about the −1/k-determinant?

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12

ML_n = M

1≤i1,...,in≤n

C · x_i₁₁. . . x_i_n_n

=

½

f(X) ∈ P(Mat_n) ; f(X) is multilinear in columns

¾ .

¨ ¥

§ ¦

U(gl_n) · det⁽⁻^1/k)(X)

=



f(X) ∈ ML_n; #I > k =⇒ X

σ∈Sn(I)

f(X · σ) = 0





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Wreath determinant

For A = (a₁, . . . ,a_n) ∈ Mat_m,n,

A^[k] = A ⊗ (

z }| {k

1, . . . ,1) = (

z }| {k

a₁, . . . ,a₁, . . . ,

z }|k {

a_n, . . . ,a_n) ∈ Mat_m,kn

Example. If A =







a₁ b₁ a₂ b₂ a₃ b₃





 ∈ Mat_3,2,

A^[3] =



a₁ a₁ a₁ b₁ b₁ b₁ a₂ a₂ a₂ b₂ b₂ b₂ a₃ a₃ a₃ b₃ b₃ b₃



 ∈ Mat_3,6 .

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14 For A ∈ Mat_kn,n, the k-wreath determinant of A is

¨ ¥

§ ¦

wrdet_k(A) = det⁽⁻^1/k)(A^[k])

= X

σ∈Skn

µ

−1 k

¶ν(σ) Yn i=1

Yk j=1

a_σ((i₋_1)k+j),i

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• By the “−1/k-alternating property” of det⁽⁻^1/k), wrdet_k(a₁, . . . ,a_i₋₁,a_i + ca_j,a_i+1, . . . ,a_n)

= wrdet_k(a₁, . . . ,a_i₋₁,a_i,a_i+1, . . . ,a_n)

• By the column-multilinearity of det⁽⁻^1/k), wrdet_k(a₁, . . . ,a_i₋₁, ca_i,a_i+1, . . . ,a_n)

= c^k wrdet_k(a₁, . . . ,a_i₋₁,a_i,a_i+1, . . . ,a_n)

¨ ¥

§ ¦

For A ∈ Mat_kn,n and P ∈ GL_n(C),

wrdet_k(AP) = det(P)^k wrdet_k(A)

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16

¨ ¥

§ ¦

For A ∈ Mat_kn,n and g ∈ S_k o S_n = Sⁿ_k o S_n, wrdet_k(g · A) = χ_n,k(g)^k wrdet_k(A) where

χ_n,k(g) = sgnτ, g = (σ, τ) ∈ Sⁿ_k oS_n We regard Sⁿ_k o S_n ⊂ S_kn by

σ((i − 1)k +j) = (i −1)k + σ_i(j), τ((i −1)k + j) = (τ(i) − 1)k + j

(1 ≤ i ≤ n, 1 ≤ j ≤ k) for σ = (σ₁, . . . , σ_n) ∈ Sⁿ_k and τ ∈ S_n.

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Determinant expression of wrdet

_k

Introduce a GL_kn × GL_n-module structure on P(Mat_kn,n) by ((g, h).f)(A) := f(^tgAh)

(g ∈ GL_kn, h ∈ GL_n, A ∈ Mat_kn,n)

We have the multiplicity-free decomposition P(Mat_kn,n) ∼= M

`(λ)≤n

M^λkn £ M^λn

by (GL_kn, GL_n)-duality.

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18 Look at the det-eigenspace for the diagonal torus T = T^kn ∼=

(C^×)^kn of GL_kn:

P(Mat_kn,n)^T^,det ∼= M

`(λ)≤n

¡M^λkn

¢T,det

£ M^λn

where

V ^T^,det = {v ∈ V ; t.v = (dett)v, t ∈ T}

• ¡

M^λkn

¢_T,det

becomes a S_kn-module (S_kn is the normalizer of T in GL_kn).

• It is known that ¡

M^λkn

¢_T,det

is irreducible S_kn-module corresponding to λ if λ ` kn.

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Let M_n,k ⊂ P(Mat_n) be the irreducible GL_kn ×GL_n-submodule corresponding to (kⁿ):

M_n,k ∼= M^(kknⁿ⁾ £ M^(kn ⁿ⁾.

As a S_kn-module, M_n,k^T^,det is irreducible (corresponding to (kⁿ)) since dimM^(kⁿ ⁿ⁾ = 1.

In particular, dimM_n,k^T^,det = f^(kⁿ⁾. Since

wrdet_k(tAP) = (dett)(detP)^k wrdet_k(A) (t ∈ T, P ∈ GL_n), it follows that wrdet_k(X) ∈ M_n,k^T^,det.

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20 Let T = (t_ij)₁_≤_i_≤_n

1≤j≤k

be a standard tableau of (kⁿ).

• For A ∈ Mat_kn,n,

det_T(A) :=

Yk p=1

det(a_t_ip_,j)₁_≤_i,j_≤_n

• Let I(T) ∈ Mat_kn,n be a matrix whose t_ij-th row vector is (0, . . . ,0,^i-th1 ,0, . . . ,0)

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Example. If n = 3, k = 2 and T =

1 2 3 5 4 6

,

det_T(A) =

¯¯¯¯

¯¯

a₁₁ a₁₂ a₁₃ a₃₁ a₃₂ a₃₃ a₄₁ a₄₂ a₄₃

¯¯¯¯

¯¯

¯¯¯¯

¯¯

a₂₁ a₂₂ a₂₃ a₅₁ a₅₂ a₅₃ a₆₁ a₆₂ a₆₃

¯¯¯¯

¯¯

I(T) =







1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1







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22

• Since

det_T(tAP) = (dett)(detP)^k det_T(A) (t ∈ T, P ∈ GL_n), it follows that det_T(X) ∈ M_n,k^T^,det.

• For standard tableaux T, U of (kⁿ),

det_T(I(U)) = δ_{T ,U}

In particular, {det_T(X)}^T are linearly independent.

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Since dimM_n,k^T^,det = f^(kⁿ⁾,

M_n,k^T^,det = M

T

C· det_T(X) Thus we have

¨ ¥

§ ¦

wrdet_k(X) = X

T

wrdet_k(I(T)) det_T(X)

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24

Example. The standard tableaux of (2³) are U1 =

1 2 3 4 5 6

, U2 =

1 2 3 5 4 6

, U3 =

1 3 2 4 5 6

, U4 =

1 3 2 5 4 6

, U5 =

1 4 2 5 3 6

,

and the corresponding matrices I(U_p) are 0

BB BB BB

@

1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1

1 CC CC CC A ,

0 BB BB BB

@

1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1

1 CC CC CC A ,

0 BB BB BB

@

1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1

1 CC CC CC A ,

0 BB BB BB

@

1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1

1 CC CC CC A ,

0 BB BB BB

@

1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1

1 CC CC CC A .

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The 2-wreath determinants wrdet₂(I(U_p)) are wrdet₂(I(U₁)) = 1

8,

wrdet₂(I(U₂)) = wrdet₂(I(U₃)) = − 1 16, wrdet₂(I(U₄)) = wrdet₂(I(U₅)) = 1

32. Hence, for A ∈ Mat_6,3,

wrdet₂(A) = 1

8 det_U₁(A) − 1

16 det_U₂(A) − 1

16 det_U₃(A) + 1

32 det_U₄(A) + 1

32 det_U₅(A).

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By the Frobenius reciprocity, 26 dim

³

M_n,k^T^,det

´Sⁿ_k

= K_(kⁿ_),(kⁿ₎ = 1.

Since wrdet_k(X) is Sⁿ_k-invariant, we have

³

M_n,k^T^,det

´Sⁿ_k

= C ·wrdet_k(X).

Consequently, by determining the proportional constant,

¨ ¥

§ ¦

wrdet_k(X) = 1 k^kn

X

σ∈Sⁿ_k

det_T₀(σ · X)

where T₀ is a standard tableau of (kⁿ) whose (i, j)-entry is (i − 1)k +j.

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As a corollary,

¨ ¥

§ ¦

Let P(Mat_kn,n)^χ^n,k^,det^k be the subspace consisting of the functions satisfying

f(gAP) = χ_n,k(g)^k(detP)^kf(A) for g ∈ S_k o S_n, P ∈ GL_n, A ∈ Mat_kn,n. Then

P(Mat_kn,n)^χ^n,k^,det^k = C · wrdet_k(X)

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28

• The “k-ply multiplicativity”

det^(α)(AP^[k]) = (detP)^k det^(α)(A^[k]) (A ∈ Mat_kn,n, P ∈ GL_n) holds only when α = −1/k.

• Then it is natural to look at the cyclic module U(gl_kn) · det^(α)(X^[k]). It is not diﬃcult to see

¨ ¥

§ ¦

U(gl_kn) · det^(α)(X^[k]) ∼= M

λ`kn fλ(α)6=0

¡M^λkn

¢^Kλ,(kn)

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• It is much harder to describe the irreducible decomposition (i.e.

determine explicitly the ‘singular values’) of the cyclic module U(gl_n) · det^(α)(X)^l

for l > 1.

• When n = 2, the ‘singular values’ are given as roots of the Jacobi polynomials:

U(gl₂) · det^(α)(X)^l ∼= M

0≤i≤l Fi(α)6=0

M^(2l2 ⁻^i,i)

where

F_i(α) = (1 + α)^l⁻ⁱ × (Jacobi polynomial)

(K.-Matsumoto-Wakayama)

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30

Expansion of wrdet

_k

For k, n ∈ N, put R_n,k :=

n

f : [kn] → [n] ; #f⁻¹(j) = k, ∀j ∈ [n]

o . (Notice that R_n,1 = S_n)

We deﬁne the sign of f ∈ R_n,k by

sgn_n,k(f) := wrdet_k(δ_f_(i),j)₁_≤_i_≤_kn

1≤j≤n

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¨ ¥

§ ¦

wrdet_k A = X

f∈Rn,k

sgn_n,k(f) Y

i∈[kn]

a_if_(i)

= X

T

sgn_n,k(T) det_T(A)

where we regard a standard tableau T = (t_ij) of (kⁿ) as an element in R_n,k by

T : [kn] 3 t_ij 7−→ i ∈ [n]

• sgn_n,k(T) = wrdet_k I(T)

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32

Deﬁne the injection

ω : S^k_n 3 (w₁, . . . , w_k) 7−→ ³

(i −1)k + j 7→ w_j(i)

´ ∈ R_n,k

¨ ¥

§ ¦

For f ∈ R_n,k,

sgn_n,k(f) = sgn(w)

µk! k^k

¶n

#¡

f · Sⁿ_k ∩ ω(S^k_n)¢

# (f · Sⁿ_k) where w ∈ S^k_n such that ω(w) ∈ f · Sⁿ_k.

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¨ ¥

§ ¦

For f ∈ R_n,k, deﬁne P_f(x₁₁, . . . , x_nk)

:= 1

#Sⁿ_k

X

σ1,...,σn∈Sⁿ_k

Yn i=1

Yk j=1

x_f_((i₋_1)k+j),σ_i_(j). Then

#¡

f · Sⁿ_k ∩ ω(S^k_n)¢

# (f · Sⁿ_k)

= the coeﬃcient of Yn i=1

Yk j=1

x_ij in P_f(x₁₁, . . . , x_nk)

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34

Example. If n = 3, k = 2 and T = U₄ =

1 3 2 5 4 6

, then

T =

µ1 2 3 4 5 6

1 2 1 3 2 3

¶

: [6] → [3], P_T(x₁₁, . . . , x₃₂)

= 1

8(x₁₁x₂₂ + x₁₂x₂₁)(x₁₁x₃₂ +x₁₂x₃₁)(x₂₁x₃₂ + x₂₂x₃₁).

The coeﬃcient of x₁₁x₂₁x₃₁x₁₂x₂₂x₃₂ is 1 4.

If we take w = ((23),(12)) ∈ S²₃, then ω(w) = T · (34) ∈ T · S³₂ and sgnw = 1. Therefore

sgn_n,k(T) = wrdet₂(U₄) = 1 ·

µ 2!

2²

¶3

· 1

4 = 1 32.

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References

• K.-Wakayama,

Invariant theory of singular α-determinants, math.RT/0603699

• Matsumoto-Wakayama,

Alpha-determinant cyclic modules of gl_n(C), J. Lie Theory 16 (2006), 393-405.

• K.-Wakayama,

Quantum α-determinant cyclic modules of U^q(gl_n), doi:10.1016/j.jalgebra.2006.12.015