The number of independent Traces and Supertraces on the Symplectic Reflection Algebra H₁,η(Γ ≀ SN)

S.E. Konstein; I.V. Tyutin

doi:10.1080/14029251.2018.1494768

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Volume 25, Issue 3, July 2018, Pages 485 - 496

The number of independent Traces and Supertraces on the Symplectic Reflection Algebra H_1,η(Γ ≀ S_N)

Authors

S.E. Konstein

I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, 53, Leninsky Prospect Moscow, 117924, Russia,konstein@lpi.ru

I.V. Tyutin

I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, 53, Leninsky Prospect, Moscow, 117924, Russia Tomsk State Pedagogical University, Kievskaya St. 60, 634061 Tomsk, Russia,tyutin@lpi.ru

Received 28 November 2017, Accepted 27 March 2018, Available Online 6 January 2021.

DOI: 10.1080/14029251.2018.1494768 How to use a DOI?
Abstract: Symplectic reflection algebra H_1,η(G) has a T(G)-dimensional space of traces whereas, when considered as a superalgebra with a natural parity, it has an S(G)-dimensional space of supertraces. The values of T(G) and S(G) depend on the symplectic reflection group G and do not depend on the parameter η.

In this paper, the values T(G) and S(G) are explicitly calculated for the groups G = Γ ≀ S_N, where Γ is a finite subgroup of Sp(2, ℂ).
Copyright: © 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

Let V := ℂ²^N, let G ⊂ Sp(2N, ℂ) be a finite group generated by symplectic reflections. In [11], it was shown that Symplectic Reflection Algebra H_1,η(G) has T(G) independent traces, where T(G) is the number of conjugacy classes of elements without eigenvalue 1 belonging to the group G ⊂ Sp(2N) ⊂ End(V), and that the algebra H_1,η(G), considered as a superalgebra with a natural parity, has S(G) independent supertraces, where S(G) is the number of conjugacy classes of elements without eigenvalue −1 belonging to G ⊂ Sp(2N) ⊂ End(V). Hereafter, speaking about spectrum, eigenvalues and eigenvectors, the rank of an element of the group algebra ℂ[G] of the group G, etc., we have in mind the representation of the group algebra ℂ[G] in the space V. Besides, we denote all the units in groups, algebras, etc., by 1, and c · 1 by c for any number c.

Apart from a few cases, there are two families of groups generated by symplectic reflections, see [7] and also [9], [2], [5]:

Family 1): G is a complex reflection group acting on 𝔥 ⊕ 𝔥^*, where 𝔥 is the space of reflection representation. In this case, G is a direct product of several groups from the following set of Coxeter groups
An(n≥1), Bn=Cn(n≥2), Dn(n≥3), E6,E7,E8,F4,G2,H3,H4,I2(n)(n≥5,n≠6).(1.1)
Family 2): G = Γ ≀ S_N, which means here G = Γ^N ⋊ S_N acting on (ℂ²)^N, where Γ is a finite subgroup of Sp(2, ℂ).

For groups G from the set (1.1), the list of values T(G) and S(G) is given in [10].

In this work, we give the values of T(G) and S(G) for the 2nd family. Namely, we found the generating functions

t(Γ,x):=∑N=0∞T(Γ≀SN)xN,(1.2)

s(Γ,x):=∑N=0∞S(Γ≀SN)xN(1.3)

for each finite subgroup Γ ⊂ Sp(2, ℂ), see Theorem 5.1.

All needed definitions are given in the Section 2; the structure, conjugacy classes and characteristic polynomials of the groups Γ ≀ S_N are described in Section 3.

To include the case N = 0 in consideration in formulas (1.2)–(1.3), it is natural to set Γ ≀ S₀ := {E} and, since dimV = 0, to set H_1,η(Γ ≀ S₀) := ℂ[{E}], where {E} is the group containing only one element E.

Applying the definitions given in Section 2 to the algebra H_1,η(Γ ≀ S₀) we deduce that

a)
if the algebra H_1,η(Γ ≀ S₀) = ℂ[{E}] is considered as superalgebra, it has only a trivial parity π ≡ 0;
b)
the algebra H_1,η(Γ ≀ S₀) = ℂ[{E}] has 1-dimensional space of traces and 1-dimensional space of supertraces; these spaces coincide;
c)
it is natural to set T(Γ ≀ S₀) = S(Γ ≀ S₀) = 1;
d)
the algebra H_1,η(Γ ≀ S₀) = ℂ[{E}] contains two Klein operators (i.e., elements satisfying conditions (2.1)–(2.3)), namely, 1 and −1.

2. Preliminaries

2.1. Traces

Let 𝒜 be an associative superalgebra with parity π. All expressions of linear algebra are given for homogenous elements only and are supposed to be extended to inhomogeneous elements via linearity.

A linear complex-valued function str on 𝒜 is called a supertrace, if

str(fg)=(−1)π(f)π(g)str(gf) for all f,g∈𝒜.

A linear complex-valued function tr on 𝒜 is called a trace, if

tr(fg)=tr(gf) for all f,g∈𝒜.

The element K ∈ 𝒜 is called a Klein operator, if

π(K)=0,(2.1)

K2=1,(2.2)

Kf=(−1)π(f)fK, for all f∈𝒜.(2.3)

Any Klein operator, if exists, establishes an isomorphism between the space of traces on 𝒜 and the space of supertraces on 𝒜.

Namely, if f ↦ tr(f) is a trace, then f ↦tr(fK^1+π(f)) is a supertrace, and if f ↦str(f) is a supertrace, then f ↦str(fK^1+π(f)) is a trace.

2.2. Symplectic reflection group

Let V = ℂ²^N be endowed with a non-degenerate anti-symmetric Sp(2N)-invariant bilinear form ω(·,·), let the vectors e_i ∈ V, where i = 1 ... , 2N, constitute a basis in V.

The matrix (ω_ij) := ω(e_i, e_j) is anti-symmetric and non-degenerate.

Let xⁱ be the coordinates of x ∈ V, i.e., x = e_i xⁱ. Then ω(x, y) = ω_ijxⁱy^j for any x, y ∈ V. The indices i are lowered and raised by means of the forms (ω_ij) and (ω^ij), where ωijωkj=δik.

Definition 2.1.

The element R ∈ Sp(2N)⊂EndV is called a symplectic reflection, if rank(R−1)=2.

Definition 2.2.

Any finite subgroup G of Sp(2N) generated by a set of symplectic reflections is called a symplectic reflection group.

In what follows, G stands for a symplectic reflection group, and ℛ stands for the set of all symplectic reflections in G.

Let R ∈ ℛ. Set

VR:=Im(R−1),(2.4)

ZR:=Ker(R−1).(2.5)

Clearly, V_R and Z_R are symplectically perpendicular, i.e., ω(V_R, Z_R) = 0, and V = V_R ⊕ Z_R.

So, let x = x_{V_R} + x_{Z_R} for any x ∈ V, where x_{V_R} ∈ V_R and x_{Z_R} ∈ Z_R. Set

ωR(x,y):=ω(xVR,yVR).(2.6)

2.3. Symplectic reflection algebra (following [3])

Let ℂ[G] be the group algebra of G, i.e., the set of all linear combinations ∑g∈Gαgg¯, where α_g ∈ ℂ.

If we were rigorists, we would write g¯ to distinguish g considered as an element of G ⊂ End(V) from the same element g¯∈ℂ[G] considered as an element of the group algebra. The addition in ℂ[G] is defined as follows:

∑g∈Gαgg¯+∑g∈Gβgg¯=∑g∈G(αg+βg)g¯

and the multiplication is defined by setting g1¯g2¯=g1g2¯. In what follows, however, we abuse notation and omit the bar sign over elements of the group algebra.

Let η be a function on ℛ, i.e., a set of constants η_R with R ∈ ℛ such that η_R₁ = η_R₂, if R₁ and R₂ belong to one conjugacy class of G.

Definition 2.3.

The algebra H_t,η(G), where t ∈ ℂ, is an associative algebra with unit 1; it is the algebra ℂ[V] of (noncommutative) polynomials in the elements of V with coefficients in the group algebra ℂ[G] subject to the relations

gx=g(x)g for any g∈G and x∈V, where g(x)=eigijxi for x=eixi,(2.7)

[x,y]=tω(x,y)+∑R∈ℛηRωR(x,y)R for any x,y∈V.(2.8)

The algebra H_t,η(G) is called a symplectic reflection algebra, see [3].

The commutation relations (2.8) suggest to define the parity π by setting:

π(x)=1, π(g)=0 for any x∈V, and g∈G,(2.9)

enabling one to consider H_t,η(G) as an associative superalgebra.

We consider the case t ≠ 0 only, for any such t it is equivalent to the case t = 1.

Let 𝒜 and ℬ be superalgebras such that 𝒜 is a ℬ-module. We say that the superalgebra 𝒜 * ℬ is a crossed product of 𝒜 and ℬ, if 𝒜 * ℬ = 𝒜 ⊗ ℬ as a superspace and

(a1⊗b1)*(a2⊗b2)=a1b1(a2)⊗b1b2,

see [14]. The element b₁(a₂) may include a sign factor imposed by the Sign Rule, see [1], p. 45.

The (super)algebra H_1,η(G) is a deform of the crossed product of the Weyl algebra W_N and the group algebra of a finite subgroup G ⊂ Sp(2N) generated by symplectic reflections.

2.4. The number of independent traces and supertraces on the symplectic reflection algebras

Theorem 2.1 ([11]).

Let the symplectic reflection group G ⊂ End(V) have T_G conjugacy classes without eigenvalue 1 and S_G conjugacy classes without eigenvalue −1.

Then the algebra H_1,η(G) has T(G) = T_G independent traces whereas H_1,η(G) considered as a superalgebra, see (2.9), has S(G) = S_G independent supertraces.

Proposition 2.1.

Let G₁ ⊂ End(V₁), G₂ ⊂ End(V₂) and G = G₁ × G₂ ⊂ End(V₁ ⊕ V₂) be symplectic reflection groups. Then T(G) = T(G₁)T(G₂) and S(G) = S(G₁)S(G₂).

Proof follows from evident relations T_G = T_G₁T_G₂, S_G = S_G₁S_G₂ and Theorem 2.1.

Proposition 2.2.

If there exists a K ∈ G such that K|_V = −1, then K is a Klein operator.

3. The group Γ ≀ S_N

3.1. Finite subgroups of Sp(2,ℂ)

The complete list of the finite subgroups Γ ⊂ Sp(2, ℂ) is as follows, see, e.g., [15]:

Γ	Order	Presence of −1	The number of conjugacy classes C(Γ)
Cyclic group Z_n := 𝕑/n𝕑	n	yes, if n is even; no, if n is odd	n
Binary dihedral group 𝒟_n	4n	yes	n + 3
Binary tetrahedral group 𝒯	24	yes	7
Binary octahedral group 𝒪	48	yes	8
Binary icosahedral group ℐ	120	yes	9

It is easy to see that each of these groups, except Z_2k+1, has C(Γ) − 1 conjugacy classes without +1 in the spectrum and has C(Γ) − 1 conjugacy classes without −1 in the spectrum. The group Z_2k+1 has C(Z_2k+1) − 1 conjugacy classes without +1 in the spectrum and it has C(Z_2k+1) conjugacy classes without −1 in the spectrum.

3.2. Symplectic reflections in Γ ≀ S_N (following [4])

Let V = ℂ²^N and let the symplectic form ω have the shape

ω:=(ϖϖ⋱ϖ), where ϖ=(01−10).(3.1)

The elements of the group Γ≀S_N have the form of N × N block matrix with 2×2 blocks. Consider the following elements of Γ ≀ S_N

(Dg,i)kl:={g,if k=l=i,1,if k=l≠i,0,otherwise,(3.2)

(Kij)kl:={δkl,if k,l≠i,k,l≠j,δkiδlj+δkjδli,otherwise,(3.3)

Sg,ij:=Dg,iDg−1,jKij,(3.4)

where i, j = 1,...,N, i ≠ j, 1 ≠ g ∈ Γ. It is clear that K_ij = K_ji and S_g,ij = S_g⁻¹,ji.

The complete set of symplectic reflections in Γ ≀ S_N consists of D_g,i, K_ij and S_g,ij, where 1 ⩽ i < j ⩽N and 1 ≠ g ∈ Γ. This set generates the group Γ ≀ S_N.

The symplectic reflections K_ij and S_g,ij lie in one conjugacy class for all i ≠ j and g ≠ 1; the elements D_g,i (g ≠ 1) and D_h,j (h ≠ 1) lie in one conjugacy class, if g and h are conjugate in Γ. So, the algebra H_1,η(Γ ≀ S_N) depends on C(Γ) parameters η, if N ⩾ 2, and on C(Γ) − 1 parameters, if N = 1. Here C(Γ) is the number of conjugacy classes in Γ including the class {1}.

3.3. Conjugacy classes (following [13])

Further, the elements of the group Γ ≀ S_N can be represented in the form Dσ where D ∈ Γ^N is a diagonal N × N block matrix, each block being a 2 × 2-matrix, and σ is N × N block matrix of permutation each block being a 2 × 2-matrix.

The product has the form:

(D1σ1)(D2σ3)=D3σ3

where σ₃ = σ₁σ₂ and D3=D1σ1D2σ1−1.

Fix an element g₀ = D₀σ₀. Since the permutation σ₀ is a product of cycles, there exists a permutation σ′ such that

σ′σ0(σ′)−1=(c1c2⋱cs), where ck are the cycles of length Lk,∑kLk=N,(3.5)

ck=(0100⋯00010⋯00001⋯0⋮⋮⋮⋱⋱⋮0000⋯11000⋯0).(3.6)

The element σ′D₀σ₀(σ′)⁻¹ has the form

σ′D0σ0(σ′)−1=(D1c1D2c2⋱Dscs),

where D_k is an L_k × L_k diagonal block matrix, each block being a 2 × 2-matrix:

Dk=(g1kg2k⋱gLkk), gik∈Γ.

Next, consider diagonal block matrices Hk=diag(h1k,h2k,…,hLkk) and the elements

HkDkckHk−1=(h1kg1k(h2k)−1h2kg2k(h3k)−1h3kg3k(h4k)−1⋱hLkkgLkk(h1k)−1)ck.

For any element h1k∈Γ, one can choose

h2k=h1kg1k, h3k=h2kg2k,…,hLkk=hLk−1kgLk−1k

such that

HkDkckHk−1=(111⋱h1kg1kg2k⋯gLkk(h1k)−1)ck.

So, each conjugacy class of Γ ≀ S_N is described by the set of cycles in the decomposition (3.5), (3.6) of σ₀, where each cycle is marked by some conjugacy class of Γ.

The cycle of length r marked by the conjugacy class α of Γ with representative g_α ∈ Γ has the shape:

Aα,r:=(111⋯gα)⋅cr=(0100…00010…00001…0………………0000…1gα000…0),(3.7)

where α = 1,...,C(Γ) and r = 1, 2,...; the matrix A_α,r and the cycle c^r are the r × r block matrices, each block being a 2 × 2 matrix.

So, each element g ∈ Γ ≀ S_N is conjugate to the element of the shape

(Aα1,r1Aα2,r2⋯Aαs,rs)(3.8)

It is convenient to describe the conjugacy class of Γ ≀ S_N with representative (3.8) by the set of nonnegative integers prα, where r = 1, 2, 3,..., and α = 1, ... , C(Γ), such that

∑α,rrprα=N.(3.9)

The value prα for some conjugacy class is the number of cycles A_α,r of length r in the decomposition (3.8) marked by the conjugacy class α of the group Γ.

Note that in [13] the notation m_r(α) is used instead of prα we use in this paper.

The restriction (3.9) can be omitted and can serve as definition of N for each set of the numbers prα.

The number of conjugacy classes in Γ ≀ S_N is equal to

C(Γ≀SN)=∑prα:∑α,rrprα=N1.

The generating function c(Γ, x) of the number of conjugacy classes is defined as

c(Γ,x):=∑N=0∞C(Γ≀SN)xN

and is equal to

c(Γ,x)=∑prαx∑α,rrprα=∑prα=0∞∏r=1∞∏α=1C(Γ)(xr)prα=∏r=1∞∏α=1C(Γ)11−xr=(Ψ(x))C(Γ),

where Ψ(x) is the Euler function

Ψ(x):=∏r=1∞11−xr

3.4. Characteristic polynomials of conjugacy classes

Before seeking the generating functions t(Γ, x) and s(Γ, x), let us find the characteristic polynomial of the conjugacy class g of Γ ≀ S_N identified by the set prα.

Let P_M(λ) := det(M − λ) be the characteristic polynomial of the matrix M. Then it is easy to see that

PAα,r(λ)=det(Aα,r−λ)=det(gα−λr)=Pgα(λr),

where the marked cycle A_α,r is defined by (3.7).

Let g ∈ Γ ≀ S_N be defined by Eq. (3.8).

Now, it is easy to show that

Pg(λ)=det(g−λ)=∏i=1sdet(Aαi,ri−λ)=∏α,r:prα⩾1det(Aα,r−λ)prα=∏α,r:prα⩾1(det(gα−λr))prα,(3.10)

if g is a representative of the congugacy class in Γ ≀ S_N corresponding to the set prα.

Definition 3.1.

We call a conjugacy class t-admissible, if its representative g ∈ Γ ≀ S_N is such that P_g(1) ≠ 0.

Definition 3.2.

We call a conjugacy class s-admissible, if its representative g ∈ Γ ≀ S_N is such that P_g(−1) ≠ 0.

Definition 3.3.

We call a marked cycle A_α,r, see Eq. (3.7), t-admissible, if P_{A_α,r}(1) ≠ 0.

Definition 3.4.

We call a marked cycle A_α,r, see Eq. (3.7), s-admissible, if P_{A_α,r}(−1) ≠ 0.

Equation (3.10) implies the following statements:

Proposition 3.1.

The conjugacy class of Γ ≀ S_N identified by the set prα is t-admissible, if and only if the marked cycle A_α,r is t-admissible for any pair α, r such that prα≠0,

Proposition 3.2.

The conjugacy class of Γ ≀ S_N identified by the set prα is s-admissible, if and only if the marked cycle A_α,r is s-admissible for any pair α, r such that prα≠0,

Recall that g_α ∈ Γ ⊂ Sp(2, ℂ), where Γ is a finite group. So detg_α = 1 and the Jordan normal form of g_α is diagonal. This implies that if g_α has +1 in its spectrum, then g_α = 1 and if g_α has −1 in its spectrum, then g_α = −1. These facts together with Eq. (3.10) imply, in their turn, the following two propositions:

Proposition 3.3.

The conjugacy class of Γ ≀ S_N identified by the set prα is t-admissible if and only if g_α ≠ 1 for all α, r with prα≠0.

Proposition 3.4.

The conjugacy class of Γ ≀ S_N identified by the set prα is s-admissible if and only if for any pair α, r such that prα≠0 , at least one of the next three conditions holds:

a)
g_α ≠ −1 and g_α ≠ 1,
b)
r is even and g_α = −1,
c)
r is odd and g_α = 1.

Note that the three sets of pairs (r, α) defined by the cases a), b), c) in Proposition 3.4 have empty pair-wise intersections.

Definition 3.5.

Let t_r(Γ) for r = 1, 2 ... be equal to the number of different α such that A_α,r is t-admissible.

Evidently,

tr(Γ)=C(Γ)−1.(3.11)

Definition 3.6.

Let s_r(Γ) for r = 1, 2 ... be equal to the number of different α such that A_α,r is s-admissible.

Evidently, if Γ ∋ −1, then

sr(Γ)=C(Γ)−1.(3.12)

and if Γ ∌ −1, then

sr(Γ)={C(Γ)−1,if r is even,C(Γ),if r is odd.(3.13)

4. Combinatorial problem

Consider the following combinatorial problem (analogous problems are considered in [8]).

Suppose we have an unlimited supply of 1-gram colored weights for each of n₁ different colors, an unlimited supply of 2-gram colored weights for each of n₂ different colors, an unlimited supply of 3-gram colored weights for each of n₃ different colors, and so on. Let an1,…,nk,…N be the number of opportunities to choose weights from our set of total mass N grams.

The problem is to find generating function

Fn1,…,nk,…(x):=∑N=0∞an1,…,nk,…NxN.

This problem is exactly the problem we discussed earlier. Namely, now we say “r-gram weight” instead of cycle of length r, and “the number of different colors n_r” instead of the number t_r (3.11) or s_r (3.12)–(3.13) of different α.

Proposition 4.1.

Fn1+m1,n2+m2,…,nk+mk,…(x)=Fn1,n2,…,nk,…(x)⋅Fm1,m2,…,mk,…(x).

Proof.

To prove this proposition, it suffices to note that

an1+m1,n2+m2,…,nk+mk,…N=∑M=0Nan1,n2,…,nk,…M⋅am1,m2,…,mk,…N−M.

Introduce the functions

fi:=Fn1i,n2i,…,nki,…, where nki=δki.

Then

fi(x)=1+xi+x2i+x3i+⋯=11−xi.

The next theorem follows from Proposition 4.1

Theorem 4.1.

Fn1,n2,…,nk,…=∏i=1∞(fi)ni.

The function F_{1, 1, 1,...} = Ψ(x) is the well-known Euler function, the generating function of the number of partitions of N into the sum of positive integers.

5. Generating functions t(Γ) and s(Γ)

Theorem 5.1.

Set

Ψ(x)=∏i=1∞11−xi (Euler function),(5.1)

Φ(x)=∏k=0∞11−x2k+1.(5.2)

Let T (Γ ≀ S_N) be the dimension of the space of traces on H_1,η(Γ ≀ S_N) and let S(Γ ≀ S_N) be the dimension of the space of supertraces on H_1,η(Γ ≀ S_N) considered as a superalgebra.

Let

t(Γ,x):=∑N=0∞T(Γ≀SN)xN and s(Γ,x):=∑N=0∞S(Γ≀SN)xN.

Then

t(Γ,x)=(Ψ(x))C(Γ)−1,s(Γ,x)=(Ψ(x))C(Γ)−1, if Γ≠Z2k+1,s(Γ,x)=(Ψ(x))C(Γ)−1Φ(x), if Γ=Z2k+1.

Proof.

To prove Theorem 5.1. we apply Theorem 4.1 to the numbers (3.11)–(3.13) of admissible conjugacy classes. It is clear that

t(Γ)=Ft1(Γ),t2(Γ),t3(Γ),…=ΨC(Γ)−1,s(Γ)=Fs1(Γ),s2(Γ),s3(Γ),…={ΨC(Γ)−1,if Γ∋−1,ΨC(Γ)−1Φ,if Γ∌−1.

Observe that Φ(x)=∑i=0∞ONxN, where O_N is the number of partitions of N into the sum of odd positive integers, and O_N coinsides with the number of independent supertraces on H_1η(S_N), see [12].

5.1. Inequality theorem

Theorem 5.2.

Let G = Γ ≀ S_N. For each positive integer N, the following statements hold:

S(G)>0,S(G)⩾T(G),S(G)=T(G) if and only if H1,η(G) contains a Klein operator.

Literally the same statements were proved for the groups G from Family 1) in [10], and hence these statements hold for the direct product of any finite number of groups from Family 1) and Family 2) defined on page 1.

Proof.

Let Γ ≠ Z_2k+1. Since each finite group Γ ∈ Sp(2N, ℂ), except Γ = Z_2k+1, contains −1, the group Γ ≀ S_N contains Klein operator K=∏i=1ND−1,i.

There is no Klein operator in H_1,η(Z_2k+1 ≀ S_N) since for this algebra, S(Z_2k+1 ≀ S_N) > T (Z_2k+1 ≀ S_N), as it follows from Theorem 5.1.

Acknowledgments

The authors (S.K. and I.T.) are grateful to Russian Fund for Basic Research (grant No. 17-02-00317) for partial support of this work.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 25 - 3
Pages: 485 - 496
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
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TY  - JOUR
AU  - S.E. Konstein
AU  - I.V. Tyutin
PY  - 2021
DA  - 2021/01/06
TI  - The number of independent Traces and Supertraces on the Symplectic Reflection Algebra H₁,η(Γ ≀ SN)
JO  - Journal of Nonlinear Mathematical Physics
SP  - 485
EP  - 496
VL  - 25
IS  - 3
SN  - 1776-0852
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DO  - 10.1080/14029251.2018.1494768
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Journal of Nonlinear Mathematical Physics

The number of independent Traces and Supertraces on the Symplectic Reflection Algebra H1,η(Γ ≀ SN)

1. Introduction

2. Preliminaries

2.1. Traces

2.2. Symplectic reflection group

Definition 2.1.

Definition 2.2.

2.3. Symplectic reflection algebra (following [3])

Definition 2.3.

2.4. The number of independent traces and supertraces on the symplectic reflection algebras

Theorem 2.1 ([11]).

Proposition 2.1.

Proposition 2.2.

3. The group Γ ≀ SN

3.1. Finite subgroups of Sp(2,ℂ)

3.2. Symplectic reflections in Γ ≀ SN (following [4])

3.3. Conjugacy classes (following [13])

3.4. Characteristic polynomials of conjugacy classes

Definition 3.1.

Definition 3.2.

Definition 3.3.

Definition 3.4.

Proposition 3.1.

Proposition 3.2.

Proposition 3.3.

Proposition 3.4.

Definition 3.5.

Definition 3.6.

4. Combinatorial problem

Proposition 4.1.

Proof.

Theorem 4.1.

5. Generating functions t(Γ) and s(Γ)

Theorem 5.1.

Proof.

5.1. Inequality theorem

Theorem 5.2.

Proof.

Acknowledgments

References

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The number of independent Traces and Supertraces on the Symplectic Reflection Algebra H_1,η(Γ ≀ S_N)

3. The group Γ ≀ S_N

3.2. Symplectic reflections in Γ ≀ S_N (following [4])