Abstract
We classify the finite-dimensional irreducible representations of the super Yangian associated with the orthosymplectic Lie superalgebra \(\mathfrak {osp}_{2|2n}\). The classification is given in terms of the highest weights and Drinfeld polynomials. We also include an R-matrix construction of the polynomial evaluation modules over the Yangian associated with the Lie superalgebra \(\mathfrak {gl}_{m|n}\), as an appendix. This is a super-version of the well-known construction for the \(\mathfrak {gl}_{n}\) Yangian and it relies on the Schur–Sergeev duality.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Data Availability
All data is available within the article.
References
Arnaudon, D., Avan, J., Crampé, N., Frappat, L., Ragoucy, E.: R-matrix presentation for super-Yangians y(osp(m|2n)). J. Math. Phys. 44, 302–308 (2003)
Arnaudon, D., Molev, A., Ragoucy, E.: On the R-matrix realization of Yangians and their representations. Ann. Henri Poincaré, 7, 1269–1325 (2006)
Berele, A., Regev, A.: Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras. Adv. Math. 64, 118–175 (1987)
Cheng, S. -J., Wang, W.: Dualities and Representations of Lie Superalgebras Graduate Studies in Mathematics, vol. 144. AMS, Providence (2012)
Cherednik, I. V.: A new interpretation of Gelfand–Tzetlin bases. Duke Math. J. 54, 563–577 (1987)
Drinfeld, V. G.: A new realization of Yangians and quantized affine algebras. Soviet Math. Dokl. 36, 212–216 (1988)
Jucys, A.: On the Young operators of the symmetric group. Lietuvos Fizikos Rinkinys 6, 163–180 (1966)
Jucys, A.: Factorization of Young projection operators for the symmetric group. Lietuvos Fizikos Rinkinys 11, 5–10 (1971)
Kac, V.: Representations of Classical Lie Superalgebras. In: Differential Geometrical Methods in Mathematical Physics. II (Proceedings. Conference, University of Bonn, Bonn, 1977), Lecture Notes in Mathematics, 676, pp 597–626. Springer, Berlin (1978)
Lu, K.: A note on odd reflections of super Yangian and Bethe ansatz. arXiv:2111.10655
Lu, K., Mukhin, E.: Jacobi–Trudi Identity and drinfeld functor for super yangian. IMRN 21, 16751–16810 (2021)
Molev, A.: Yangians and Classical Lie Algebras Mathematical Surveys and Monographs, vol. 143. AMS, Providence (2007)
Molev, A.: Representations of the Yangians associated with Lie superalgebras \(\mathfrak {osp}(1|2n)\). arXiv:2109.023612109.02361
Molev, A.: Odd reflections in the Yangian associated with \(\mathfrak {gl}(m|n)\). Lett. Math. Phys., 112(1). Paper No. 8, 15 pp (2022)
Murphy, G.E.: A new construction of Young’s seminormal representation of the symmetric group. J. Algebra 69, 287–291 (1981)
Nazarov, M. L.: Quantum Berezinian and the classical Capelli identity. Lett. Math Phys. 21, 123–131 (1991)
Nazarov, M.: Yangians and Capelli identities. In: Olshanski, G. I. (ed.) Kirillov’s Seminar on Representation Theory. American Mathematical Society Translations, vol. 181, pp 139–163. American Mathematical Society, Providence (1998)
Nazarov, M.: Representations of twisted Yangians associated with skew Young diagrams. Sel. Math. (N.S.) 10, 71–129 (2004)
Nazarov, M.: Yangian of the general linear Lie superalgebra. SIGMA 16, 112 (2020). 24 pages
Sergeev, A. N.: Tensor algebra of the identity representation as a module over the Lie superalgebras \(\mathfrak {gl}(n,m)\) and Q(n). Mat. Sb. (N.S.) 123(165), 422–430 (1984)
Zamolodchikov, A. B., Zamolodchikov, Al. B.: Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models. Ann. Phys. 120, 253–291 (1979)
Zhang, R. B.: Representations of super Yangian. J. Math. Phys. 36, 3854–3865 (1995)
Zhang, R. B.: The \(\mathfrak {gl}(m|n)\) super Yangian and its finite-dimensional representations. Lett. Math. Phys. 37, 419–434 (1996)
Funding
Open Access funding enabled and organized by CAUL and its Member Institutions
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
This work was supported by the Australian Research Council, grant DP180101825. The author has no competing interests to declare that are relevant to the content of this article.
Additional information
Presented by: Michela Varagnolo
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A: Polynomial Evaluation Modules over the Yangian \(\mathrm {Y}(\mathfrak {gl}_{m|n})\)
Appendix A: Polynomial Evaluation Modules over the Yangian \(\mathrm {Y}(\mathfrak {gl}_{m|n})\)
1.1 A.1 Defining Relations and Representations
Given nonnegative integers m and n, we will use the notation \(\bar {\imath }=0\) for \(i=1,\dots ,m\) and \(\bar {\imath }=1\) for \(i=m+1,\dots ,m+n\). Introduce the \(\mathbb {Z}_{2}\)-graded vector space \(\mathbb {C}^{m|n}\) over \(\mathbb {C}\) with the basis \(e_{1},e_{2},\dots ,e_{m+n}\), where the parity of the basis vector ei is defined to be \(\bar {\imath }\mod 2\). Accordingly, equip the endomorphism algebra \({\text {End} }\mathbb {C}^{m|n}\) with the \(\mathbb {Z}_{2}\)-gradation, where the parity of the matrix unit eij is found by \(\bar {\imath }+\bar {\jmath }\mod 2\).
A standard basis of the general linear Lie superalgebra \(\mathfrak {gl}_{m|n}\) is formed by elements Eij of the parity \(\bar {\imath }+\bar {\jmath }\mod 2\) for \(1\leqslant i,j\leqslant m+n\) with the commutation relations
The Yang R-matrix associated with \(\mathfrak {gl}_{m|n}\) is the rational function in u given by
where P is the permutation operator,
Following [16] and [19], define the Yangian \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) as the \(\mathbb {Z}_{2}\)-graded algebra with generators \(t_{ij}^{(r)}\) of parity \(\bar {\imath }+\bar {\jmath }\mod 2\), where \(1\leqslant i,j\leqslant m+n\) and \(r=1,2,\dots \), satisfying the quadratic relations
written in terms of the formal series
Combining them into the matrix T(u) = [tij(u)] and regarding it as the element
we can write the defining relations in the standard RTT-form (2.4) with th e R-matrix (A.1).
The universal enveloping algebra \( \mathrm {U}(\mathfrak {gl}_{m|n})\) can be regarded as a subalgebra of \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) via the embedding \( E_{ij}\mapsto t_{ij}^{(1)}(-1)^{\bar {\imath }}, \) while the mapping
defines the evaluation homomorphism \(\text {ev}{:} \mathrm {Y}(\mathfrak {gl}_{m|n})\to \mathrm {U}(\mathfrak {gl}_{m|n})\).
The Yangian \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) is a Hopf algebra with the coproduct defined by
A representation V of the algebra \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) 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 and the (m + n)-tuple \(\pi (u)=(\pi _{1}(u),\dots ,\pi _{m+n}(u))\) is called its highest weight.
Given an arbitrary tuple \(\pi (u)=(\pi _{1}(u),\dots ,\pi _{m+n}(u))\) of formal series of the form (A.4), the Verma moduleM(π(u)) is defined as the quotient of the algebra \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) by the left ideal generated by all coefficients of the series tij(u) with \(1\leqslant i<j\leqslant m+n\), and tii(u) − πi(u) for \(i=1,\dots ,m+n\). We will denote by L(π(u)) its irreducible quotient. The isomorphism class of L(π(u)) is determined by π(u). Necessary and sufficient conditions on π(u) for the representation L(π(u)) to be finite-dimensional are known due to [23]. Their extension to arbitrary parity sequences via odd reflections was given in [14]; see also [10].
Consider the irreducible highest weight representation V (π) of the Lie superalgebra \(\mathfrak {gl}_{m|n}\) with the highest weight \(\pi =(\pi _{1},\dots ,\pi _{m+n})\), associated with the standard Borel subalgebra. This means that Eiiζ = πiζ for \(i=1,\dots ,m+n\) and Eijζ = 0 for i < j, where ζ is the highest vector of V (π). The representation V (π) is finite-dimensional if and only if the highest weight π satisfies the conditions
see [9]. Use the evaluation homomorphism (A.2) to equip V (π) with a \( \mathrm {Y}(\mathfrak {gl}_{m|n})\)-module structure. The evaluation module V (π) is isomorphic to the Yangian highest weight module L(π(u)), where the components of the highest weight π(u) are
1.2 A.2 Schur–Sergeev Duality and Fusion Procedure
We will follow [4, Sec. 3.2] to recall a version of the Schur–Sergeev duality going back to [3] and [20]. An (m,n)-hook partition \(\lambda =(\lambda _{1},\lambda _{2},\dots )\) is a partition with the property \(\lambda _{m+1}\leqslant n\). This means that the Young diagram λ is contained in the (m,n)-hook as depicted below. The figure also illustrates the partitions \(\mu =(\mu _{1},\dots ,\mu _{m})\) and \(\nu =(\nu _{1},\dots ,\nu _{n})\) associated with λ. They are introduced by setting
and
where \(\lambda ^{\prime }\) denotes the conjugate partition so that \(\lambda ^{\prime }_{j}\) is the length of column j in the diagram λ:
![](http://media.springernature.com/lw287/springer-static/image/art%3A10.1007%2Fs10468-022-10121-w/MediaObjects/10468_2022_10121_Figa_HTML.png)
We will associate two (m + n)-tuples of nonnegative integers with λ by
The tensor product space \((\mathbb {C}^{m|n})^{\otimes d}\) is naturally a module over both \(\mathfrak {gl}_{m|n}\) and the symmetric group \(\mathfrak {S}_{d}\). For the action of the basis elements of \(\mathfrak {gl}_{m|n}\) we have
while the transposition \((a b)\in \mathfrak {S}_{d}\) with a < b acts by (ab)↦Pab with
The images of \( \mathrm {U}(\mathfrak {gl}_{m|n})\) and \(\mathbb {C}\mathfrak {S}_{d}\) in the endomorphism algebra \({\text {End} }(\mathbb {C}^{m|n})^{\otimes d}\) satisfy the double centralizer property, which leads to the multiplicity-free decomposition
summed over the (m,n)-hook partitions λ with d boxes, where Sλ is the Specht module over \(\mathfrak {S}_{d}\) associated with λ. By representing the group algebra \(\mathbb {C}\mathfrak {S}_{d}\) as the direct sum of matrix algebras
we can think of Sλ as the canonical irreducible module over \({\text {Mat}}_{f_{\lambda }}(\mathbb {C})\) isomorphic to \(\mathbb {C}^{f_{\lambda }}\). The diagonal matrix units \(e_{\mathcal {U}}=e^{\lambda }_{\mathcal {U}\mathcal {U}}\in {\text {Mat}}_{f_{\lambda }}(\mathbb {C})\) parameterized by standard λ-tableaux \(\mathcal {U}\) are primitive idempotents in \(\mathbb {C}\mathfrak {S}_{d}\). We may conclude that if λ is an (m,n)-hook partition with d boxes, then the image \(e_{\mathcal {U}}(\mathbb {C}^{m|n})^{\otimes d}\) is isomorphic to the \(\mathfrak {gl}_{m|n}\)-module V (λ♯). If λ is not contained in the (m,n)-hook, then the image is zero.
Explicit formulas for the idempotents \(e_{\mathcal {U}}\) can be derived with the use of the orthonormal Young basis of Sλ via the Jucys–Murphy elements \(x_{1},\dots ,x_{d}\) of the group algebra \(\mathbb {C}\mathfrak {S}_{d}\) defined by
Given a standard λ-tableau \(\mathcal {U}\), denote by \(\mathcal {V}\) the standard tableau obtained from \(\mathcal {U}\) by removing the box α occupied by d. Then the shape of \(\mathcal {V}\) is a diagram which we denote by λ−. A box outside λ− is called addable, if the union of λ− and the box is a Young diagram. We let c = j − i denote the content of the box α = (i,j) and let \(a_{1},\dots ,a_{l}\) be the contents of all addable boxes of λ− except for α. The Jucys–Murphy formula gives an inductive rule for the calculation of \(e_{\mathcal {U}}\):
see [8] and [15]. The idempotents \(e_{\mathcal {U}}\) can also be obtained from the fusion procedure for the symmetric group; see [5, 7] and [17], and we recall a version following [12, Sec. 6.4], where it was essentially derived from Eq. A.6. Take d complex variables \(u_{1},\dots ,u_{d}\) and consider the rational function with values in \(\mathbb {C}\mathfrak {S}_{d}\) defined by
where the product is taken in the lexicographical order on the set of pairs (a,b). Suppose that λ ⊩ d and let \(\mathcal {U}\) be a standard λ-tableau. Let \(c_{a}=c_{a}(\mathcal {U})\) for \(a=1,\dots ,d\) be the contents of \(\mathcal {U}\) so that ca = j − i if a occupies the box (i,j) in \(\mathcal {U}\). Then the consecutive evaluations of the rational function \(\phi (u_{1},\dots ,u_{d})\) are well-defined and the value coincides with the primitive idempotent \(e_{\mathcal {U}}\) multiplied by the product of hook lengths h(λ) of λ,
1.3 A.3 Yangian Action on Polynomial Modules
The Yang R-matrix (A.1) is a solution of the Yang–Baxter equation
in \({\text {End} }(\mathbb {C}^{m|n})^{\otimes 3}\) with Rab(u) = 1 − Pabu− 1. This implies that the mapping T(u)↦R(u) defines a representation of the algebra \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) on the space \(\mathbb {C}^{m|n}\), known as the vector representation. In terms of the generating series it has the form
For an (m,n)-hook partition λ ⊩ d fix a standard λ-tableau \(\mathcal {U}\). As above, let \(c_{1},\dots ,c_{d}\) be the contents of the respective entries in \(\mathcal {U}\). By using the coproduct (A.3) and the shift automorphism of \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) defined by Eq. 2.9, we get a representation of the Yangian on the space \((\mathbb {C}^{m|n})^{\otimes d}\) defined by
which is written in terms of elements of the algebras
with the first copy of the endomorphism algebra labelled by 0.
Another form of the vector representation is related to Eq. A.9 via twisting with the super-transposition automorphism
We then get the action of the algebra \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) on the space \(\mathbb {C}^{m|n}\) given by
cf. Eq. A.2. It can be written in a matrix form as \(T(u)\mapsto R^{\prime }(-u)\) with
Accordingly, the composition of the representation (A.10) with the automorphism (A.11) yields another representation of the Yangian on the space \((\mathbb {C}^{m|n})^{\otimes d}\) given by
where \(R^{\prime }_{0 a}(u)=1-Q_{0 a} u^{-1}\) and
Similar to Eq. A.10, this representation can also be obtained by using the opposite coproduct on the Yangian, which is the composition of (A.3) and the \(\mathbb {Z}_{2}\)-graded flip operator.
Theorem 1.1
The subspace \(L_{\mathcal {U}}=e_{\mathcal {U}}(\mathbb {C}^{m|n})^{\otimes d}\) is invariant under the Yangian actions (A.10) and (A.13). Moreover, the respective representations of the Yangian \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) on \(L_{\mathcal {U}}\) are isomorphic to the highest weight representations L(π♭(u)) and L(π♯(u)), where
and
Proof
Let \(E_{\mathcal {U}}\in {\text {End} }(\mathbb {C}^{m|n})^{\otimes d}\) denote the image of the idempotent \(e_{\mathcal {U}}\) under the representation of the symmetric group on \((\mathbb {C}^{m|n})^{\otimes d}\). We have the relation
which plays a key role in the derivation of the fusion formula (A.7); cf. [17] and [12, Prop. 6.4.4]. Apply the anti-automorphism \(e_{ij}\mapsto e_{ji}(-1)^{\bar {\imath }\bar {\jmath }+\bar {\imath }}\) to the zeroth copy of the endomorphism algebra \({\text {End} }\mathbb {C}^{m|n}\) to derive
Together with Eq. A.14 this proves the first part of the theorem.
Furthermore, relation (A.15) shows that the action (A.13) on \(L_{\mathcal {U}}\) is the composition of the evaluation homomorphism (A.2) and the action (A.5) of \(\mathfrak {gl}_{m|n}\). Hence this representation is isomorphic to the evaluation module V (λ♯)≅L(π♯(u)). Similarly, relation (A.14) shows that the Yangian action (A.10) on the subspace \(L_{\mathcal {U}}\) is the composition of the evaluation homomorphism (A.2) and the action (A.5) of \(\mathfrak {gl}_{m|n}\) twisted by the automorphism
On the other hand, by [4, Sec. 2.4], an application of a chain of odd reflections shows that the extremal weight of the \(\mathfrak {gl}_{m|n}\)-module V (λ♯) with respect to the opposite Borel subalgebra is λ♭. That is, there is a nonzero vector η ∈ V (λ♯) of the weight λ♭ such that Ei,i+ 1η = 0 for all i≠m and Em+n,1η = 0. By taking the lowest vector with respect to the action of \(\mathfrak {gl}_{m}\oplus \mathfrak {gl}_{n}\) we conclude that V (λ♯) contains a nonzero vector ζ of the weight \((\mu _{m},\dots ,\mu _{1},\lambda ^{\prime }_{n},\dots ,\lambda ^{\prime }_{1})\) such that Eijζ = 0 for all i > j. Thus, the vector ζ is the highest vector of the Yangian module \(L_{\mathcal {U}}\) and its weight is found by taking into account Eqs. A.2 and A.16 so that this module is isomorphic to L(π♭(u)). □
By using the coproduct (A.3) and the vector representation (A.12) instead of (A.9), for any complex parameters za we get a representation of the Yangian on the space \((\mathbb {C}^{m|n})^{\otimes d}\) defined by
For suitable parameters za, the subspaces \(L_{\mathcal {U}}=e_{\mathcal {U}}(\mathbb {C}^{m|n})^{\otimes d}\) associated with the standard tableaux \(\mathcal {U}\) of shapes λ = (d) and λ = (1d), turn out to be invariant under this Yangian action as well. The primitive idempotents \(e_{\mathcal {U}}\) associated with the standard row and column tableaux are, respectively, the symmetrizer and anti-symmetrizer in \(\mathbb {C}\mathfrak {S}_{d}\),
Note that Eqs. A.6 and A.7 yield multiplicative formulas for h(d) and a(d).
Corollary 1.2
-
(i)
The subspace \(h^{(d)}(\mathbb {C}^{m|n})^{\otimes d}\) is invariant under the action
$$ T(u)\mapsto R^{\prime}_{01}(-u-d+1){\dots} R^{\prime}_{0d}(-u). $$Moreover, the representation of the Yangian \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) on this subspace is isomorphic to the highest weight representation with the highest weight \( \left (1+d u^{-1},1\dots ,1\right ). \)
-
(ii)
The subspace \(a^{(d)}(\mathbb {C}^{m|n})^{\otimes d}\) is invariant under the action
$$ T(u)\mapsto R^{\prime}_{01}(-u+d-1){\dots} R^{\prime}_{0d}(-u). $$Moreover, the representation of the Yangian \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) on this subspace is isomorphic to the highest weight representation with the highest weight
$$ \left( \underbrace{1+u^{-1},\dots,1+u^{-1}}_d ,1,\dots,1\right)\qquad\text{if}\quad d\leqslant m, $$(A.17)and
$$ \left( \underbrace{1+u^{-1},\dots,1+u^{-1}}_m ,1+(m-d) u^{-1},1,\dots,1\right)\qquad\text{if}\quad d> m. $$
Proof
Multiplying both sides of Eq. A.15 by the image \(P_{\omega }=P_{1, d} P_{2, d-1}\dots \) of the longest permutation \(\omega \in \mathfrak {S}_{d}\) from the left, we get
Since ωh(d) = h(d) and ωa(d) = sgnω ⋅ a(d), both parts of the corollary follow from the particular cases of Theorem A.1 concerning the action (A.13) for the row and column tableaux \(\mathcal {U}\). □
The following corollary in the case n = 0 was used in the construction of the fundamental modules over the Yangians of types B, C and D in [2, Sec. 5.3]. It was also applied to the orthosymplectic Yangians in [13] and in the proof of Proposition 2.9 in the previous section. Equip the tensor product space \((\mathbb {C}^{m|n})^{\otimes d}\) with the action of \( \mathrm {Y}(\mathfrak {gl}_{m|n})\) by setting
where the generators act in the respective copies of the vector space \(\mathbb {C}^{m|n}\) via the rule (A.12). Set
Corollary 1.3
For any \(1\leqslant d\leqslant m\) the vector ξd has the properties
and
Proof
This is immediate from Corollary A.2(ii), because the vector ξd is the highest vector of the \( \mathrm {Y}(\mathfrak {gl}_{m|n})\)-module \(a^{(d)}(\mathbb {C}^{m|n})^{\otimes d}\) with the highest weight (A.17). □
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Molev, A. Representations of the Super Yangians of Types A and C. Algebr Represent Theor 26, 1007–1027 (2023). https://doi.org/10.1007/s10468-022-10121-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-022-10121-w