Centralizer algebras of the primitive unitary reflection group of order 96
aa r X i v : . [ m a t h . R T ] M a y Centralizer algebras of the primitive unitaryreflection group of order 96
Masashi Kosuda and Manabu Oura
Abstract
Among the unitary reflection groups, the one on the title is singledout by its importance in, for example, coding theory and number theory.In this paper we start with describing the irreducible representations ofthis group and then examine the semi-simple structure of the centralizeralgebra in the tensor representation.
The group, which we denote by H , on the title of this paper consists of 96matrices of size 2 by 2. It is the unitary group generated by reflections (u.g.g.r.),numbered as No.8 in Shephard-Todd [14]. This group, as well as No.9 in thesame list, has long been recognized. The purpose of the present paper is to givea contribution to H by decomposing the centralizer algebra of H in the tensorrepresentation into irreducible components.We shall give an outline of the first statement in Abstract. The group H naturally acts on the polynomial ring C [ x, y ] of 2 variables over the complexnumber field C , i.e. Af ( x, y ) = f ( ax + by, cx + dy ) , A = (cid:18) a bc d (cid:19) ∈ H for f ∈ C [ x, y ]. We consider the invariant ring C [ x, y ] H = { f ∈ C [ x, y ] : Af = f, ∀ A ∈ H } of H . This ring has a rather simple structure. It is generated by two alge-braically independent homogeneous polynomials of degrees 8 and 12, and con-versely this nature characterizes the u.g.g.r. Brou´e-Enguehard [6] found a map Keywords: Centralizer algebra, unitary reflection group, Bratteli diagramMSC2010: Primary 05E10, Secondary 05E05 05E15 05E18.running head: Centralizer algebra of H in the tensor representation f ( x, y ) of degree n from the invariant ring. Introducing theta constants θ ab ( τ ) = X m ∈ Z exp 2 πi (cid:20) τ (cid:16) m + a (cid:17) + (cid:16) m + a (cid:17) b (cid:21) , we get a modular form f ( θ (2 τ ) , θ (2 τ )) of weight n/ SL (2 , Z ). Moreoverthis map is an isomorphism from the invariant ring of H onto the ring ofmodular forms for SL (2 , Z ).Next we proceed to coding theory. Let F = { , } be the field of twoelements and F n the vector space of dimension n over F equipped with theusual inner product ( u, v ) = u v + · · · + u n v n . The weight of a vector u isthe number of non-zero coordinates of u . A code of length n is by definition alinear subspace of F n . We impose two conditions on codes. The first one is theself-duality which says that a code C coincides with its dual code C ⊥ , that is, C = C ⊥ in which C ⊥ = { u ∈ F n : ( u, v ) = 0 , ∀ v ∈ C } . The second one is the doubly-evenness which means wt ( u ) ≡ , ∀ u ∈ C. These two notions give rise to the relation with invariant theory via the weightenumerator W C ( x, y ) = X v ∈ C x n − wt ( v ) y wt ( v ) of a code C . In fact, if C is self-dual, we have W C (( x − y ) / √ , ( x + y ) / √
2) = W C ( x, y )and if C is doubly even, we have W C ( x, iy ) = W C ( x, y ) . We mention that a self-dual and doubly even code of length n exists if and onlyif n is a multiple of 8.Now we can state the connections among all what we have mentioned. Takea positive integer n ≡ n is an invariant of H and W C ( θ (2 τ ) , θ (2 τ ))is a modular form of weight n/ SL (2 , Z ). Gleason [9] showed that theinvariants of degree n can be spanned by the weight enumerators of self-dualdoubly even codes of length n . Finally any modular form of weight n/ . The whole theory with more general results could be found in [11], [12] fromwhich our notation H comes.Besides the importance of H , the motivation of this paper could be foundin Brauer [5], Weyl [16]. One of the main ingredients there is the commuta-tor algebra where invariant theory comes into play. We follow Weyl. Givenany group of linear transformations in an n -dimensional space. Take covariantvectors y (1) , . . . , y ( f ) and contravariant vectors ξ (1) , . . . , ξ ( f ) . A linear trans-formation acts on covariant vectors cogrediently and on contravariant vectors contragradiently . Then the matrices k b ( i · · · i f ; k · · · k f ) k in the tensor spaceobtained from the invariants X i ; k b ( i · · · i f ; k · · · k f ) ξ (1) i · · · ξ ( f ) i f y (1) k · · · y ( f ) k f form the commutator algebra of H in the tensor representation. The problemhere is to decompose this algebra into simple parts. It is quite natural to applythis philosophy to our group H as we will in this paper ( cf . [1]). H H which yieldsthe character table. At the end of this section we discuss invariant theory of H under the irreducible representations.The unitary reflection group H is a finite group in U generated by thefollowing matrices T and D : T = 1 + i (cid:18) − (cid:19) = 1 √ (cid:18) ǫ ǫǫ ǫ (cid:19) , D = (cid:18) i (cid:19) . Here ǫ = exp(2 πi/ H is 96 and it has 16conjugacy classes C , . . . , C . Each conjugacy class has the following represen-tative: C ∋ (cid:18) (cid:19) , C ∋ T = √ (cid:18) ǫ ǫǫ ǫ (cid:19) , C ∋ T = (cid:18) i i (cid:19) , C ∋ T = √ (cid:18) ǫ ǫ ǫ ǫ (cid:19) , C ∋ T = (cid:18) − − (cid:19) , C ∋ T = (cid:18) − i − i (cid:19) , C ∋ D = (cid:18) i (cid:19) , C ∋ DT = √ (cid:18) ǫ ǫǫ ǫ (cid:19) , C ∋ DT = (cid:18) i − (cid:19) , C ∋ DT = √ (cid:18) ǫ ǫ ǫ ǫ (cid:19) , C ∋ DT = (cid:18) − − i (cid:19) , C ∋ DT = √ (cid:18) ǫ ǫ ǫ ǫ (cid:19) , C ∋ DT = (cid:18) − i
00 1 (cid:19) , C ∋ DT = √ (cid:18) ǫ ǫ ǫ ǫ (cid:19) , C ∋ D = (cid:18) − (cid:19) , C ∋ D T = (cid:18) i − i (cid:19) . H . In the following, we construct all of them one by one.First we note that any group has the trivial representation which maps eachelement of the group to 1. We denote that of H by ( ρ , V ). The determinantwhich maps T and D to − i and i respectively also gives a one-dimensionalirreducible representation. We call it ( ρ , V ). The tensor product ρ ⊗ also givesa one-dimensional irreducible representation, which maps both T and D to − ρ , V ). Also ρ ⊗ ρ defines a one-dimensional representation. Wename it ( ρ , V ).Next we consider two-dimensional representations. The natural representa-tion ( ρ , V ) which maps T and D to the defining matrices above is irreducible,since neither of one-dimensional D -invariant subspaces are T -invariant. Takingtensor products with the one-dimensional representations above and the naturalrepresentation, we have further 3 two-dimensional irreducible representations, ρ = ρ ⊗ ρ , ρ = ρ ⊗ ρ and ρ = ρ ⊗ ρ . There are 2 more two-dimensionalirreducible representations which we will deal with later.As a subrepresentation of ρ ⊗ ρ , we have a three-dimensional irreduciblerepresentation. Let h e , e i be a basis of V which gives the natural represen-tation. Then h e ⊗ e , e ⊗ e , e ⊗ e , e ⊗ e i gives a basis for the tensorrepresentation ρ ⊗ . With respect to this basis, the representation matrices of T and D are ρ ⊗ ( T ) = i − −
11 1 − − − − and ρ ⊗ ( D ) = diag(1 , i, i, − . If we put e ′ = e ⊗ e , e ′ = e ⊗ e + e ⊗ e and e ′ = e ⊗ e , then h e ′ , e ′ , e ′ i isobviously a D -invariant subspace. It is easy to check that it is also T -invariant.Hence it gives a three-dimensional representation. We name it ( ρ , V ). Therepresentation matrices with respect to this basis are ρ ( T ) = i − − and ρ ( D ) = diag(1 , i, − . Since each one-dimensional D -invariant subspace of V is not T -invariant, therepresentation ( ρ , V ) is irreducible. Similarly to the previous case, we havefurther 3 three-dimensional irreducible representations, ρ = ρ ⊗ ρ , ρ = ρ ⊗ ρ and ρ = ρ ⊗ ρ .Next we look for a four-dimensional irreducible representation in ( ρ ⊗ ρ , V ⊗ V ). Let h e i ⊗ e ′ j | i = 1 , , j = 1 , , i be a basis of V ⊗ V (lexicographical order). Then we have the following representation matrices of4 and D : ρ ⊗ ρ ( T ) = − i − − − − − − −
11 0 − − − − − ,ρ ⊗ ρ ( D ) = diag(1 , i, − , i, − , − i ) . If we put e ′′ = e ⊗ e ′ , e ′′ = e ⊗ e ′ + e ⊗ e ′ , e ′′ = e ⊗ e ′ + e ⊗ e ′ , and e ′′ = e ⊗ e ′ , according to the eigen values of ρ ⊗ ρ ( D ), then h e ′′ k | k = 1 , , , i isobviously a D -invariant subspace. It is also easy to check that it is T -invariant.Hence it gives a four-dimensional representation. We name it ( ρ , V ). Therepresentation matrices with respect to this basis are ρ ( T ) = − i − − − − − − and ρ ( D ) = diag(1 , i, − , − i ) . As we saw in the previous case, none of one-dimensional D -invariant subspaces of V is T -invariant. Now consider two-dimensional D -invariant subspaces. Sinceall eigen spaces of ρ ( D ) are one-dimensional, we find that a two-dimensional D -invariant subspace is of the form h e ′′ i , e ′′ j i ( i = j ). Let W be h e ′′ , e ′′ i and takea non-zero vector v = a e ′′ + b e ′′ from W . Then we have ρ ( T ) v = − i a + 3 b ) e ′′ + ( a + b ) e ′′ + ( a − b ) e ′′ + ( a − b ) e ′′ ] . In order that ρ ( T ) v ∈ W , it must hold that a = b = 0. This contradictsthe assumption that v is non-zero vector. Hence we find that W is not T -invariant. Similar arguments hold for all two-dimensional D -invariant subspaces {h e ′′ i , e ′′ j i} ≤ i 11 3 3 1 − − − − 11 1 − − − − − − − − − − − − ,ρ ⊗ ρ ( D ) = diag(1 , i, − , − i, i, − , − i, . If we put e ′′′ = e ⊗ e ′′ + e ⊗ e ′′ , and e ′′′ = e ⊗ e ′′ + e ⊗ e ′′ , then h e ′′′ , e ′′′ i is T - and D -invariant subspace. We name it ( ρ , V ). The representation matriceswith respect to this basis are ρ ( T ) = − (cid:18) − (cid:19) and ρ ( D ) = diag(1 , − . Similarly to the previous ones, we can check that this representation is irre-ducible. Further ρ = ρ ⊗ ρ also defines a two-dimensional irreducible repre-sentation.So far, we have got 16 irreducible representations. Since H has 16 conjugacyclasses, { ( ρ i , V i ) } i =1 are a complete representatives of all irreducible represen-tations of H . Accordingly, the character table of H is also derived.6 C C C C C C C C C C C C C C C C T T T T T D DT DT DT DT DT DT DT D D T order 1 8 4 8 2 4 4 6 4 12 4 3 4 12 2 4 χ χ − − − − − − χ − i − i − i − i − i − i − − χ i − − i − − i i − − i i − − χ − − − − χ − − − − − χ − i − i − i i − i − i − − − i i χ i − − i − − i − i i i − − i − i χ − i − i − i − − i − i − i − i i χ i − − i i − i i − − i − − i − i χ − − − − − − χ − − − − χ i − − i − i − i i − i − χ − i − i − − i i − i i − χ − i − i − i − i χ i − − i − − i i e conclude this section with adding a few words on invariant theory of H under irreducible representations ( cf . [8]). Let ρ be one of the d -dimensionalirreducible representation of H . Then ρ ( H ) acts naturally on the polynomialring of d variables. We denote the invariant ring under this action by C [ ρ ] H .The orders of ρ i ( H ) are1 , , , | {z } dim 1 , , , , , , | {z } dim 2 , , , , | {z } dim 3 , , | {z } dim 4 . The dimension 1 case aside, the invariant rings C [ ρ ] H , C [ ρ i ] H ( i = 7 , , , , C [ ρ ] H are weighted polynomial rings. In the sense of [14], all ρ i ( H ) ( i = 7 , , , ρ ( H ) to ρ ( H ). We already know the ring C [ ρ ] H . The ring C [ ρ ] H can be generated by the polynomials of degrees 2 and3, and the ring C [ ρ ] H by those of degrees 2 , ρ ( H ) is equivalent to G (3 , , 2) and ρ ( H ) to G (2 , , ρ case has a somewhat complicated structure. The ring C [ ρ ] H is amodule of rank 32 over the polynomial ring. We note that calculations herewere done with Magma [4]. In the previous section, we have found complete representatives of all irreduciblerepresentations. In this section, we see how tensor powers of ρ are decomposedinto irreducible ones.We begin with the general theory (see for example Curtis-Reiner[7]). Let χ , . . . , χ s be the set of all irreducible characters of a finite group G . For any(not necessarily irreducible) representation ( ρ, V ) of G , let χ be its character.Then χ can be uniquely expressed a sum of irreducible characters: χ = m χ + · · · + m s χ s . Now suppose that χ has its character values ( k , . . . , k s ) on the conjugacy classes( C , . . . , C s ). Then we get( k , . . . , k s ) = ( χ ( C ) , . . . , χ ( C s ))= m ( χ ( C ) , . . . , χ s ( C s )) + · · · + m s ( χ s ( C ) , . . . , χ s ( C s ))= ( m , . . . , m s ) χ ( C ) · · · χ ( C s )... . . . ... χ s ( C ) · · · χ s ( C s ) . If we let X denote the matrix of the character table, then the above relation issimply written as ( k , . . . , k s ) = ( m , . . . , m s ) X . (1)8y the linear independence of the irreducible characters, X is non-singular.Hence we have ( m , . . . , m s ) = ( k , . . . , k s ) X − . (2)In order to examine the structure of the centralizer algebra of the tensorrepresentation, it is useful to investigate how the tensor product of the naturaland an irreducible representation is decomposed into the irreducible ones. Inthe following, we go back to our case and decompose ρ ⊗ ρ i ( i = 1 , , . . . , χ · χ = χ ,χ · χ = χ ,χ · χ = χ ,χ · χ = χ . Further, we can directly read the following from the character table: χ · χ = χ ,χ · χ = χ . Next, consider χ · χ ( C , . . . , C ). Again from the character table, we have χ · χ ( C , . . . , C )= (4 , , , , , , − , , − , , − , , − , , , . Using the identity (2), we have(0 , , , , , , , , , , , , , , , χ · χ ( C , . . . , C ) X − . This means χ · χ = χ + χ . In a similar way we have χ · χ = χ + χ ,χ · χ = χ + χ ,χ · χ = χ + χ ,χ · χ = χ + χ ,χ · χ = χ + χ ,χ · χ = χ + χ ,χ · χ = χ + χ ,χ · χ = χ + χ + χ ,χ · χ = χ + χ + χ . 9y the above calculation, we obtain the Hasse diagram of the decomposition of ρ ⊗ k into irreducible ones. ρ , = 1 ρ , ρ , + 1 = 2 ρ , ρ , + 1 = 5 ρ , ρ , ρ , ρ , + 3 + 1 + 1 = 15 ρ , ρ , ρ , + 1 + 5 = 51 ρ , ρ , ρ , ρ , ρ , + 1 + 10 + 6 + 5 = 187 ρ , ρ , ρ , 21 15 + 7 + 21 = 715 ρ , ρ , ρ , ρ , ρ , 21 15 + 7 + 28 + 36 + 21 = 2795 ρ , ρ , ρ , 85 35 + 51 + 85 = 11051 ⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❖❖❖❖❖❖❖❖❖❖❖❖❖⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄❖❖❖❖❖❖❖❖❖❖❖❖❖ ⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ From this diagram we can read off the multiplicity of each irreducible repre-sentation ( ρ i , V i ) in V ⊗ k by counting the number of paths from the top vertexindexed by ρ in the 1-st row to the corresponding vertex in the k -th row. Weput the multiplicity on the right side of each irreducible representation. Further,we calculated the square sums of the multiplicities on the each row.Let A k = End H ( V ⊗ k ) be the centralizer algebra of H in V ⊗ k , where H acts on V diagonally. By the Schur-Weyl reciprocity [13, 16], this diagram isthe Bratteli diagram of the algebra sequence C = A ⊂ A ⊂ A ⊂ · · · . (For the Bratteli diagram, see for example Goodman-de la Harpe-Jones [10], § k -th row is thedimension of End H ( V ⊗ k ). We will examine it in detail in the next section.10 Centralizer algebra In the previous section, we have seen that the dimensions of A k = End H ( V ⊗ k )( k = 1 , , . . . ) are 1, 2, 5, 15, 51, 187, 715,. . . . According to “The On-LineEncyclopedia of Integer Sequences”[15], these terms coincide with the fist fewterms of the expression (3 · k − + 2 k − + 1) / 3. This is indeed the case. Inorder to prove this, we calculate the size of each simple component of A k .Let d ( i ) j be the multiplicity of ρ i in the tensor representation ρ j , whichcoincides with the size of the corresponding simple component of A j . By theBratteli diagram of A j given in the previous section, we have the recursiveformulae as follows. First note that the irreducible representations ρ , ρ , ρ of H again appear in the bottom of the diagram, as well as the 5-th row of thediagram. This implies that the diagram periodically grows up as k increases.The iteration is as follows:[ ρ , ρ , ρ ] → [ ρ , ρ , ρ , ρ , ρ ] → [ ρ , ρ , ρ ] → [ ρ , ρ , ρ , ρ , ρ ] → · · · . Hence based on the Bratteli diagram of the 9-th row from the 5-th row, we canobtain the following recursive formulae: d (4)4 ℓ +2 = d (8)4 ℓ +1 ,d (3)4 ℓ +2 = d (10)4 ℓ +1 ,d (14)4 ℓ +2 = d (8)4 ℓ +1 + d (16)4 ℓ +1 ,d (13)4 ℓ +2 = d (10)4 ℓ +1 + d (16)4 ℓ +1 ,d (6)4 ℓ +2 = d (16)4 ℓ +1 , d (9)4 ℓ +3 = d (4)4 ℓ +2 + d (14)4 ℓ +2 ,d (7)4 ℓ +3 = d (3)4 ℓ +2 + d (13)4 ℓ +2 ,d (15)4 ℓ +3 = d (14)4 ℓ +2 + d (13)4 ℓ +2 + d (6)4 ℓ +2 , d (1)4( ℓ +1) = d (9)4 ℓ +3 ,d (2)4( ℓ +1) = d (7)4 ℓ +3 ,d (11)4( ℓ +1) = d (7)4 ℓ +3 + d (15)4 ℓ +3 ,d (12)4( ℓ +1) = d (9)4 ℓ +3 + d (15)4 ℓ +3 ,d (5)4( ℓ +1) = d (15)4 ℓ +3 , d (8)4( ℓ +1)+1 = d (2)4( ℓ +1) + d (11)4( ℓ +1) ,d (10)4( ℓ +1)+1 = d (1)4( ℓ +1) + d (12)4( ℓ +1) ,d (16)4( ℓ +1)+1 = d (11)4( ℓ +1) + d (12)4( ℓ +1) + d (5)4( ℓ +1) . Note also that if we allow the possibility d ( i ) j = 0, the recursions above are still11alid even for the 1-st to 4-th row. Hence we have the following: d (8)4 ℓ +1 = d (2)4 ℓ + d (11)4 ℓ = d (7)4 ℓ − + ( d (7)4 ℓ − + d (15)4 ℓ − )= 2 d (7)4 ℓ − + d (15)4 ℓ − = 2( d (3)4 ℓ − + d (13)4 ℓ − ) + ( d (14)4 ℓ − + d (13)4 ℓ − + d (6)4 ℓ − )= 2 d (3)4 ℓ − + d (14)4 ℓ − + 3 d (13)4 ℓ − + d (6)4 ℓ − = 2 d (10)4 ℓ − + ( d (8)4 ℓ − + d (16)4 ℓ − ) + 3( d (10)4 ℓ − + d (16)4 ℓ − ) + d (16)4 ℓ − = d (8)4( ℓ − + 5 d (10)4( ℓ − + 5 d (16)4( ℓ − ( ℓ > . (3)Similarly we have d (10)4 ℓ +1 = 5 d (8)4( ℓ − + d (10)4( ℓ − + 5 d (16)4( ℓ − (4)and d (16)4 ℓ +1 = 5 d (8)4( ℓ − + 5 d (10)4( ℓ − + 11 d (16)4( ℓ − (5)From the recursion (3), (4) and (5), and the initial condition ( d (8)1 , d (10)1 , d (16)1 ) =(0 , , d (8)4 ℓ +1 = − ( − ℓ ℓ ,d (10)4 ℓ +1 = ( − ℓ ℓ ,d (16)4 ℓ +1 = − 13 + 16 ℓ . By the initial recursion formulae, we immediately obtain d (4)4 ℓ +2 = − ( − ℓ ℓ ,d (3)4 ℓ +2 = ( − ℓ ℓ ,d (14)4 ℓ +2 = − ( − ℓ ℓ ,d (13)4 ℓ +2 = ( − ℓ ℓ ,d (6)4 ℓ +2 = − 13 + 16 ℓ , (9)4 ℓ +3 = − ( − ℓ + 13 + 2 · ℓ ,d (7)4 ℓ +3 = ( − ℓ + 13 + 2 · ℓ ,d (15)4 ℓ +3 = − 13 + 4 · ℓ d (1)4( ℓ +1) = − ( − ℓ + 13 + 2 · ℓ ,d (2)4( ℓ +1) = ( − ℓ + 13 + 2 · ℓ ,d (11)4( ℓ +1) = ( − ℓ + 2 · ℓ ,d (12)4( ℓ +1) = − ( − ℓ + 2 · ℓ ,d (5)4( ℓ +1) = − 13 + 4 · ℓ . Thus we have obtained the size of each simple component of A k . If we applysimple considerations to the order of simple components, the sizes are uniformlydescribed as follows. Theorm 4.1. Let A k = End H ( V ⊗ k ) be a centralizer algebra of H in V ⊗ k ,where H acts on V diagonally. Then A k has the following multi-matrix struc-ture. A k ∼ = M d + ( k ) ( C ) ⊕ M d − ( k ) ( C ) ⊕ M d ( k ) ( C ) if k = 2 m − ,M d + ( k ) ( C ) ⊕ M d − ( k ) ( C ) ⊕ M d ( k ) ( C ) ⊕ M e + ( k ) ( C ) ⊕ M e − ( k ) ( C ) if k = 2 m, where d ± ( k ) = ± m − + 13 + 2 · m − ,d ( k ) = − 13 + 4 m − and e ± = ± m − + 2 · m − . Calculating the square sum of the dimensions of the simple components of A k in cases k = 2 m − k = 2 m , we finally obtain the following corollaryas we expected. Corollary 4.2. dim A k = 2 k − + 2 k − . H could be described in terms of the symmetric polynomials in 4noncommuting variables [2] and/or the universal embedding of the symplecticdual polar space DS p (2 k, 2) [3]. It would be interesting that these points becomeclear. Acknowledgment . This work was supported by JSPS KAKENHI GrantNumber 25400014. References [1] Bannai, E., Bannai, E., Ozeki, M. and Teranishi, Y., On the ring of simulta-neous invariants for the Gleason-MacWilliams group, European J. Combin. 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E., The Symmetric Group: Representations, Combinatorial Al-gorithms, and Symmetric Functions , Second Edition, Graduate Text inMathematics 203, Springer-Verlag, New York, 2001.[14] Shephard, G. C. and Todd, J. A., Finite unitary reflection groups, CanadianJ. Math. (1954), 274-304.[15] Sloane, N. J. A. editor, The On-Line Encyclopedia of Integer Sequences,(https://oeis.org).[16] Weyl, H., The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N. J., 1939. Department of Mathematical Sciences, University of the Ryukyus,Okinawa, 903-0213, JAPAN E-mail address : [email protected] Graduate School of Natural Science and Technology, KanazawaUniversity, Ishikawa, 920-1192 Japan