Representations of the super-Yangian of type $B(n,m)$

We are concerned with finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras ${\frak osp}_{2n+1|2m}$. Every such representation is highest weight and we use embedding theorems and odd reflections of Yangian type to derive necessary conditions for an irreducible highest weight representation to be finite-dimensional. We conjecture that these conditions are also sufficient. We prove the conjecture in the case $n=1$ and arbitrary $m\geqslant 1$.


Introduction
We continue the investigation of finite-dimensional irreducible representations of the orthosymplectic Yangians initiated in [8] and [9], where classification theorems were proved in the particular cases of osp 1|2m and osp 2|2m .We consider the Yangians associated with the Lie superalgebras osp 2n+1|2m which form the series B(n, m) in the Kac classification of simple Lie superalgebras.
We will work with the extended Yangian X(osp 2n+1|2m ) and use its R-matrix definition as given by Arnaudon et al. [1].By a standard argument, every finite-dimensional irreducible representation of X(osp 2n+1|2m ) is a highest weight representation.It is isomorphic to the irreducible quotient L(λ(u)) of the Verma module M(λ(u)) associated with a tuple λ(u) = λ 1 (u), . . ., λ m (u), λ m+1 (u), . . ., λ m+n+1 (u) (1.1) The tuple is called the highest weight of the representation.Both the Yangian Y(gl n|m ) associated with the general linear Lie superalgebra gl n|m and the extended Yangian X(o 2n+1 ) associated with the orthogonal Lie algebra o 2n+1 can be regarded as subalgebras of the orthosymplectic Yangian via embeddings Y(gl n|m ) ֒→ X(osp 2n+1|2m ) and X(o 2n+1 ) ֒→ X(osp 2n+1|2m ). (1.2) Hence, by considering the cyclic spans of the highest vector of L(λ(u)) with respect to these subalgebras, and using the finite-dimensionality conditions for highest weight representations of Y(gl n|m ) from [13] and of X(o 2n+1 ) from [2], we get certain necessary conditions for the representation L(λ(u)) to be finite-dimensional.
We will give a detailed proof of Theorem 1.1 in Sec.3.1.Then we bring some evidence provided by particular cases to support the following conjecture.
Conjecture 1.2.The conditions on the highest weight λ(u) stated in Theorem 1.1 are sufficient for the representation L(λ(u)) to be finite-dimensional.Conjecture 1.2 holds for n = 0 due to the results of [8], where conditions (1.8) and (1.9) do not occur.We will prove the following theorem in Sec. 5. Before specializing to the case n = 1, we give necessary and sufficient conditions for the representations L(λ(u)) of X(osp 2n+1|2m ) with linear highest weights to be finite-dimensional (Theorem 4.1).They will show that Conjecture 1.2 holds in this case.
We then use the linear highest weight modules to prove Conjecture 1.2 in the case of generic highest weights (Corollary 4.5).
A key role in the proof of Theorem 4.1 will be played by the polynomial representations of X(osp 2n+1|2m ) which are produced in a way similar to the Yangian Y(gl n|m ), as reviewed e.g. in [9,Appendix].Moreover, the proof will also use a special fundamental representation of the Lie superalgebra osp 2n+1|2m which admits an extension to the Yangian, as shown in Lemma 4.3.This work was completed during the first author's visit to the Laboratoire d'Annecy-le-Vieux de Physique Théorique.He thanks the lab for the support and hospitality.

Definitions and preliminaries
For given positive integers m and n consider the parity sequences s = s 1 . . .s m+n of length m+n, where each term s i is 0 or 1, and the total number of zeros is n.For notational convenience we also set s m+n+1 = 0 and regard this as an additional entry of s which will not vary.The standard sequence s st = 1 . . . 10 . . .0 is defined by s i = 1 for i = 1, . . ., m and s i = 0 for i = m + 1, . . ., m + n.
Suppose a parity sequence s is fixed.We will simply write ī to denote its i-th term s i .Introduce the involution i → i ′ = 2n + 2m − i + 2 on the set {1, 2, . . ., 2n + 2m + 1} and set i ′ = ī for i = 1, . . ., m + n + 1.Consider the Z 2 -graded vector space C 2n+1|2m over C with the basis e 1 , e 2 , . . ., e 1 ′ , where the parity of the basis vector e i is defined to be ī mod 2. Accordingly, equip the endomorphism algebra End C 2n+1|2m with the Z 2 -gradation, where the parity of the matrix unit e ij is found by ī +  mod 2.
We will consider even square matrices with entries in Z 2 -graded algebras, their (i, j) entries will have the parity ī +  mod 2. The algebra of even matrices over a superalgebra A will be identified with the tensor product algebra End C 2n+1|2m ⊗ A, so that a matrix A = [a ij ] is regarded as the element We will use the involutive matrix super-transposition t defined by (A t ) ij = a j ′ i ′ (−1) ī+ θ i θ j , where for i = 1, . . ., 1 ′ we set otherwise.
We will also regard t as the linear map In the case of multiple tensor products of the endomorphism algebras, we will indicate by t a the map (2.1) acting on the a-th copy of End C 2n+1|2m .A standard basis of the general linear Lie superalgebra gl 2n+1|2m is formed by elements E ij of the parity ī +  mod 2 for 1 i, j 1 ′ with the commutation relations We will regard the orthosymplectic Lie superalgebra osp 2n+1|2m as the subalgebra of gl 2n+1|2m spanned by the elements Introduce the permutation operator P by and set The R-matrix associated with osp 2n+1|2m is the rational function in u given by This is a super-version of the R-matrix originally found in [12].Following [1], we define the extended Yangian X(osp 2n+1|2m ) = X(osp s 2n+1|2m ) corresponding to the parity sequence s as a Z 2 -graded algebra with generators t (r) ij of parity ī +  mod 2, where 1 i, j 1 ′ and r = 1, 2, . . ., satisfying the following defining relations.Introduce the formal series and combine them into the matrix T (u) = [t ij (u)].Consider the elements of the tensor product algebra End The defining relations for the algebra X(osp 2n+1|2m ) take the form of the RT T -relation The algebra X(osp s 2n+1|2m ) associated with an arbitrary parity sequence s is isomorphic to the Yangian X(osp 2n+1|2m ) associated with the standard sequence s st ; an isomorphism is given by the map t ij (u) → t σ(i),σ(j) (u) for a suitable permutation σ of the set {1, . . ., 1 ′ }.
The product T (u − κ) T t (u) is a scalar matrix with where c(u) is a series in u −1 .Moreover, all the coefficients of the series c(u) belong to the center ZX(osp 2n+1|2m ) of X(osp 2n+1|2m ) and generate the center.The Yangian Y(osp 2n+1|2m ) is defined as the subalgebra of X(osp 2n+1|2m ) which consists of the elements stable under the automorphisms for all series f (u) We have the tensor product decomposition The Yangian Y(osp 2n+1|2m ) can also be regarded as the quotient of X(osp 2n+1|2m ) by the relation c(u) = 1.
Explicitly, the defining relations (2.3) can be written with the use of super-commutator in terms of the series (2.2) as follows: Each of the mappings and defines an automorphism of X(osp 2n+1|2m ).The universal enveloping algebra U(osp 2n+1|2m ) can be regarded as a subalgebra of X(osp 2n+1|2m ) via the embedding This fact relies on the Poincaré-Birkhoff-Witt theorem for the orthosymplectic Yangian which was pointed out in [1] and a detailed proof is given in [4]; cf.[2,Sec. 3].It states that the algebra X(osp 2n+1|2m ) is generated by the coefficients of the series c(u) and t ij (u) with the conditions Moreover, given any total ordering on the set of the generators, the ordered monomials with the powers of odd generators not exceeding 1, form a basis of the algebra.For a given parity sequence s = 0s ′ which begins with 0 consider the extended Yangian X(osp s ′ 2n−1|2m ) and for the parity sequence s = 1s ′ beginning with 1 consider the extended Yangian X(osp s ′ 2n+1|2m−2 ).In both cases, let the indices of the generators t (r) ij of these algebras range over the sets 2 i, j 2 ′ and r = 1, 2, . . . .The following embedding properties were proved in [10, Thm 3.1] for a standard parity sequence, and the proof expends to arbitrary sequences s without any significant changes.The mapping defines the injective homomorphisms in the cases s = 0s ′ and s = 1s ′ , respectively.The extended Yangian X(osp 2n+1|2m ) is a Hopf algebra with the coproduct defined by The image of the series c(u) is found by the relation ∆ : c(u) → c(u) ⊗ c(u) and so the Yangian Y(osp 2n+1|2m ) inherits the Hopf algebra structure from X(osp 2n+1|2m ).

Highest weight representations
We will keep a parity sequence s fixed.A representation V of the algebra X(osp s 2n+1|2m ) is called a highest weight representation if there exists a nonzero vector ξ ∈ V such that V is generated by ξ, for some formal series The vector ξ is called the highest vector of V .
Proposition 3.1.The series λ i (u) associated with a highest weight representation V satisfy the consistency conditions for i = 1, . . ., m + n.Moreover, the coefficients of the series c(u) act in the representation V as the multiplications by scalars determined by c(u As Proposition 3.1 shows, the series λ i (u) in (3.1) with i > m+n+1 are uniquely determined by the first m+n+1 series.We will call the corresponding tuple λ(u) = (λ 1 (u), . . ., λ m+n+1 (u)) the highest weight of V .

Proof of Theorem 1.1
As we outlined in the Introduction, we will use the embeddings (1.2).Recall from [11] that the Yangian Y(gl n|m ) (corresponding to the standard parity sequence s st ) is defined as the Z 2 -graded algebra with generators t (r) ij of parity ī +  mod 2, where 1 i, j m + n and r = 1, 2, . . ., while The defining relations can be written in terms of the generating series and they have the form The first embedding in (1.2) is given by Since the representation L(λ(u)) is finite-dimensional, then so is the Y(gl n|m )-module Y(gl n|m )ξ defined via this embedding.This is a highest weight representation with the highest weight (λ 1 (u), . . ., λ m+n (u)).Hence, conditions (1.7) and (1.8) (the latter excluding j = m + n) must hold due to the classification theorem of [13] (see also its review in [7] in the context of odd reflections).
The second embedding in (1.2) was given in [10,Cor. 3.2].The same argument as with the first embedding shows that since the cyclic span X(o 2n+1 )ξ is a finite-dimensional highest weight representation with the highest weight (λ m+1 (u), . . ., λ m+n+1 (u)), conditions (1.8) must hold due to the classification theorem of [2] in the case of the Yangian X(o 2n+1 ).
Conditions (1.7) and (1.8) imply that by twisting the representation L(λ(u)) by a suitable automorphism (2.5), if necessary, we may assume that all components λ i (u) of the highest weight λ(u) are polynomials in u −1 ; that is, for some p ∈ Z + we have with λ (r) i ∈ C for r = 1, . . ., p. Consider the parity sequence s obtained from s st by replacing the subsequence s m s m+1 = 10 with 01.To calculate the highest weight of the module L(λ(u)) associated with s, apply the corresponding odd reflection by using [7,Thm 4.4].We will assume that the parameters λ (r)  m and λ (r) m+1 are numbered in such a way that m+1 for all 1 r, s k.Then set Note that these series coincide with those given by (1.4).
Although the arguments of [7] deal with representations of Y(gl n|m ), the odd reflections apply to representations of the Yangian X(osp 2n+1|2m ) as well, by taking into account the first embedding in (1.2).The final step in the proof of Theorem 4.4 therein should just be adjusted to use the re-labelling automorphism of X(osp 2n+1|2m ) acting on the generators by where σ ∈ S 2n+2m+1 is the product of the transpositions (m, m + 1) and ((m + 1) ′ , m ′ ).

Representations with linear highest weights
Here we will be concerned with the representations L(λ(u)) of the Yangian X(osp 2n+1|2m ) (associated with the standard parity sequence s st ), such that all components of the highest weight λ(u) are linear in u −1 as given in (1.10).Observe that the twist of L(λ(u)) by the composition of the automorphism (2.5) for f (u) = u/(u + λ m+n+1 ) and the automorphism (2.8) for a = −λ m+n+1 yields a representation with the linear highest weight, where the components are changed by the rule λ i → λ i − λ m+n+1 .Therefore, we will not restrict generality by assuming λ m+n+1 = 0 so that λ(u) = (1 + λ 1 u −1 , . . ., 1 + λ m+n u −1 , 1).(4.1) The cyclic span U(osp 2n+1|2m )ξ defined via the embedding (2.10) is a highest weight representation of osp 2n+1|2m with F ij ξ = 0 for all 1 i < j 1 ′ .It follows from (2.4) that the generator t (1) m+n+1,m+n+1 belongs to the center of X(osp 2n+1|2m ) and so it acts as multiplication by 0 in L(λ(u)).Hence, F ii ξ = (−1) ī λ i ξ for i = 1, . . ., m + n by (2.10), and so the highest weight of the osp 2n+1|2m -module L = U(osp 2n+1|2m )ξ is given by (−λ 1 , . . ., −λ m , λ m+1 , . . ., λ m+n ). or for some l ∈ Z + ; if l n then the second part of condition (4.4) is understood as Proof.Suppose first that dim L(λ(u)) < ∞.Since the cyclic span L is finite-dimensional, the osp 2n+1|2m -highest weight (4.2) must satisfy the finite-dimensionality conditions for irreducible highest weight representations of osp 2n+1|2m as obtained by Kac [5]; see also [3,Sec. 2.1].This implies the desired conditions on the components of the highest weight.Conversely, suppose that the conditions on the components of the highest weight λ(u) are satisfied.Equip the set of osp 2n+1|2m -weights of L with the standard partial order determined by the choice of the positive root system corresponding to the upper-triangular subalgebra of osp 2n+1|2m .The next lemma will be our main tool for proving that certain vectors in L(λ(u)) are zero.
Since the action of the coefficients of the series t jj (u) on the vector η does not change its osp 2n+1|2m -weight, it follows from the Poincaré-Birkhoff-Witt theorem that a nonzero vector η satisfying (4.6) would generate a submodule of L(λ(u)) which does not contain ξ.However, this is impossible due to the irreducibility of L(λ(u)), thus proving that η = 0.
The proof for an arbitrary subset S ∈ {1, . . ., m + n} follows from the observation that the condition t i,i+1 (u)η = 0 for any given i ∈ {1, . . ., m + n} can be replaced with t (i+1) ′ ,i ′ (u)η = 0 for the conclusion to remain valid.This is clear from relation (2.4) written in the form

By using the expansion
and setting the degree of t (r) ij equal to r, we derive that + a linear combination of monomials of degree < r.
The claim is now verified by an easy induction on r.
We proceed by considering a particular representation which will play the role of a fundamental Yangian module.
Proof.We will be using the eigenvalues λ i ′ (u) of the diagonal operators t i ′ i ′ (u) on ξ for the values i = 1, . . ., m + n.They are found in Proposition 3.1 and given by The highest weight (4.2) associated with the given components λ i corresponds to a finitedimensional representation of the Lie superalgebra osp 2n+1|2m .Therefore, it will be sufficient to demonstrate that all elements t (r) ij of the extended Yangian with r 2 act on L as the zero operators.Furthermore, it is clear from the defining relations (2.7) that it will be enough to verify that these elements act as the zero operators on the highest vector ξ.
Observe that Indeed, by (2.7), up to a sign factor depending on j, the expression t j, j+1 (u) t j+1, j (v)ξ equals which vanishes because λ j (u) = λ j+1 (u).Furthermore, t j+1, j (v)ξ is annihilated by the remaining simple root series t i,i+1 (u) with i = j.This follows by applying (2.7) to [t i,i+1 (u), t j+1, j (v)] for i < j, and to [t j+1, j (v), t i,i+1 (u)] for i > j.Thus (4.7) follows from Lemma 4.2.By taking commutators of the left hand side of (4.7) with suitable elements t i+1,i we come to more general relations Since the transposition automorphism (2.9) preserves the module L(λ(u)), relations (4.8) extend to the values As the next step, we verify that m+1,m ξ = 0. (4.9) It is clear from (2.7) that the extression on the left hand side is annihilated by the series t i,i+1 (u) for i = 1, . . ., m − 1, m + 1, . . ., m + n.Furthermore, by (2.7) the expression The coefficient of ξ simplifies to and (4.9) follows from Lemma 4.2.Take commutators of the left hand side of (4.9) with suitable elements t (1) i+1 i , where i satisfies the conditions of (4.7), to derive that m+j,i ξ = 0 for i = 1, . . ., m and j = 1, . . ., n. (4.10) The same calculation verifies that m+n+1,m+n ξ = 0 m+n+1,m+j ξ = 0 for j = 1, . . ., n. ( The application of the automorphism (2.9) yields the respective counterparts of (4.10) and (4.11) for the transposed series.Furthermore, to derive that m+n+1,m ξ = 0 we note first that the expression on the left hand side is annihilated by the series t i,i+1 (u) for i = 1, . . ., m as verified by the same calculation as for (4.9).For i = m + 1, . . ., m + n the expression m+n+1,m ξ equals The previously verified relations (4.10) and (4.8) imply that this expression is zero.Hence, we may conclude that m+n+1,i ξ = 0 for i = 1, . . ., m + n together with the transposed relations implies by the application of the automorphism (2.9).Finally, we will show by a reverse induction that for each j = 1, . . ., m + n we have i ′ j ξ = 0 for i = 1, . . ., j. ( Taking first i = j note the relation j+1, j ξ For the standard parity sequence s st and d ∈ {1, . . ., m}, use the coproduct (2.13) to equip the tensor product space (C 2n+1|2m ) ⊗d with the action of X(osp 2n+1|2m ) by setting where the generators act in the respective copies of the vector space (C 2n+1|2m ) ⊗d via the rule (4.15).Set The calculations of [9,Appendix] show that the cyclic span X(osp 2n+1|2m ) ξ d is a highest weight representation of X(osp 2n+1|2m ) with the highest weight λ(u) whose components are found by We will denote the irreducible quotient of this representation by L ♯ d .Now consider the parity sequence s = 0 . . .01 . . . 1 and recall that the algebra X(osp s 2n+1|2m ) is isomorphic to the extended Yangian X(osp 2n+1|2m ) associated with the standard parity sequence.For d ∈ {1, . . ., n} equip the tensor product space (C 2n+1|2m ) ⊗d with the action of X(osp s 2n+1|2m ) by setting where the generators act in the respective copies of the vector space (C 2n+1|2m ) ⊗d via the rule (4.15).The vector ξ d defined by the same formula (4.16) now generates a highest weight representation of X(osp s 2n+1|2m ) with the highest weight λ(u) whose components are found by λ i (u) = 1 + u −1 for i = 1, . . ., d and λ i (u) = 1 for i = d + 1, . . ., m + n + 1.
We will denote the irreducible quotient of this representation by L ♭ d .
It is clear from the description of the Yangian odd reflections in Sec.3.1, that the linear highest weights of the form (4.1) are transformed in the same way as the highest weights (4.2) are transformed with respect to the corresponding orthosymplectic Lie superalgebra odd reflections, as described in [3,Sec. 1.3].In particular, we get the following correspondence between the linear highest weights of the modules associated with Young diagrams implied by [3,Sec. 2.4].
Recall that an (m, n)-hook partition Γ = (Γ 1 , Γ 2 , . . . ) is a partition satisfying the condition Γ m+1 n.This means that the Young diagram Γ is contained in the (m, n)-hook as depicted below.The figure also illustrates the partitions µ = (µ 1 , . . ., µ m ) and ν = (ν 1 , . . ., ν n ) associated with Γ.They are introduced by setting and where Γ ′ denotes the conjugate partition so that Γ ′ j is the length of column j in the diagram Γ: We will associate two (m + n)-tuples of integers with Γ by We are now in a position to complete the proof of the theorem.Suppose first that conditions (4.4) hold.They mean that the highest weight (4.2) coincides with Γ ♯ for certain (m, n)-hook partition Γ. Hence it will be sufficient to show that the highest module L(λ(u)) associated with Γ in this way, is finite-dimensional.
Recall the irreducible highest weight representations of L ♯ d and L ♭ d of the extended Yangian constructed above and denote by L ♯ d a and L ♭ d a their respective compositions with the shift automorphism (2.8).
Given an (m, n)-hook partition Γ, consider the tensor product module where Γ m+1 should be replaced by 0. According to Lemma 4.4, the cyclic X(osp 2n+1|2m )-span of the tensor product of the highest weight vectors of the modules L ♯ d a is a highest weight module with the highest weight given by (4.1) with As we recalled above, by applying a sequence of odd reflections, we find that the highest weight of the irreducible quotient L Γ of this cyclic span associated with the parity sequence s = 0 . . .01 . . . 1 is found as Γ ♭ , where Γ is the Young diagram with m rows Γ 1 , . . ., Γ m .
If the parameter l in (4.4) exceeds n−1, it should be understood as equal to n in the argument below; in that case we set ν n+1 := 0. Consider the tensor product module By Lemma 4.4, the cyclic X(osp s 2n+1|2m )-span of the tensor product of the highest weight vectors of the tensor factors is a highest weight module with the highest weight associated with Γ ♭ .All modules involved in the tensor products are finite-dimensional and so is the irreducible quotient of the cyclic span.This completes the proof of the sufficiency of conditions (4.4).Now suppose that conditions (4.3) hold.The argument will be quite similar to the above, taking the finite-dimensional representation L • constructed in Lemma 4.3 as the starting point.Consider the tensor product module where we set λ m+1 := −n.By Lemma 4.4, the cyclic X(osp 2n+1|2m )-span of the tensor product of the highest weight vectors of the tensor factors is a highest weight module with the highest weight given by By applying a sequence of odd reflections, we can regard the irreducible quotient L • of the cyclic span as a X(osp s 2n+1|2m )-module with s = 0 . . .01 . . . 1, whose highest weight is found by Finally, L(λ(u)), regarded as a X(osp s 2n+1|2m )-module, is isomorphic to the irreducible quotient of the cyclic span of the tensor product of the highest weight vectors of the tensor factors in where we set λ m+n+1 := 1/2.Thus, L(λ(u)) is finite-dimensional.
The following corollary confirms Conjecture 1.2 in the case of generic highest weights.
Corollary 4.5.Suppose that the components of the highest weight λ(u) are given by (3.4) with the condition that for each i = 1, . . ., m + n + 1 none of the differences λ Proof.Due to Theorem 1.1, we only need to establish the sufficiency of the conditions.They imply that because of the additional assumptions, for each i the parameters λ i can be re-numbered in such a way that each linear weight satisfies the conditions of Theorem 1.1 for a = 1, . . ., p.By Theorem 4.1, each representation L(λ (a) (u)) is finite-dimensional and so is the cyclic span of the tensor products of the highest vectors in L(λ (1) (u)) ⊗ . . .⊗ L(λ (p) (u)).
By Lemma 4.4, the irreducible quotient of the cyclic span is isomorphic to L(λ(u)), thus implying that it is finite-dimensional.
5 Classification theorem for representations of X(osp 3|2m ) In this section we specialize to the case n = 1 and prove Theorem 1.3.We will show that the conditions of Theorem 1.1 imply that the highest weight λ(u) can be split into linear weights in a way similar to the proof of Corollary 4.5.
As in the proof of Theorem 1.1, by twisting the representation L(λ(u)) by a suitable automorphism (2.5), we may assume that all components λ i (u) of the highest weight λ(u) are given by (3.4).Furthermore, we may also assume that the parameters λ (r)  m and λ (r) m+1 are numbered in such a way that relations (3.5) hold for certain k ∈ {0, 1, . . ., p}, while λ (r)  m = λ (s) m+1 for all 1 r, s k.Then the application of the odd reflection yields the subsequent formulas for λ [1]  m (u) and λ [1] m+1 (u) given therein.Suppose first that λ Finally, the case where some parameters λ It suffices to observe that conditions (1.8) for j = m + 1 and (1.9) will hold for the polynomials λ i (u)/δ(u) with i = m, m + 1 and m + 2 which allows us to complete the proof of Theorem 1.3 in this case in the same way as above.The components do satisfy the conditions of Theorem 1.1, but no splitting into linear highest weights of the form (1.10) satisfying the conditions of Theorem 1.1 is possible.Moreover, such a splitting is not possible for the highest weights corresponding to the parity sequences s = 010 and 001 either.The question whether dim L(λ(u)) < ∞ remains open for this example.

(4. 2 )Theorem 4 . 1 .
For complex numbers a and b we will write a → b or b ← a to mean that a − b ∈ Z + .The representation L(λ(u)) of X(osp 2n+1|2m ) with the highest weight (4.1) is finite-dimensional if and only if either

i
is an integer for a = b.Then the representation L(λ(u)) is finite-dimensional if and only if λ(u) satisfies the conditions of Theorem 1.1.
[3,)According to[3, Example 2.53], if the highest weight (4.2) for the standard parity sequence s st coincides with Γ ♯ for an (m, n)-hook partition Γ, then the highest weight for the parity sequence s = 0 . . .01 . . . 1 obtained by a sequence of odd reflections, coincides with Γ ♭ .