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I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$.

I can easily find the above info for orthogonal groups and unitary groups (the representations coming from certain spaces of harmonic polynomials) but cannot seem to find such a description for the groups $\text{Sp}(n)$.

In particular I am aware that the irreducible representations of $\text{Sp}(2)$ are classified by Young diagram parameters $(a,b)$ with $a\geq b \geq 0$ but I would like to compute with such spaces so need an explicit description if possible.

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    $\begingroup$ Are you talking about complex representations? Also, please make precise what you mean by "explicit description". There are many descriptions around of varying degrees of explicitness (standard monomial theory, crystal bases, LS-paths etc.). There are also a realizations as harmonic polynomials. $\endgroup$
    – Friedrich Knop
    Commented Jun 20, 2016 at 14:48
  • $\begingroup$ Yes, I wish to know about complex representations. As for explicit, I simply mean any vector space I can generate and compute the action of $\text{Sp}(2)$ on, e.g. spaces of harmonic polynomials. $\endgroup$
    – fretty
    Commented Jun 20, 2016 at 14:50
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    $\begingroup$ Sp(2) also goes by the name Spin(5). $\endgroup$
    – Theo Johnson-Freyd
    Commented Jun 20, 2016 at 14:51
  • $\begingroup$ I have seen this mentioned but could you explain what the irreducible representations of Spin(5) are? $\endgroup$
    – fretty
    Commented Jun 20, 2016 at 15:20
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    $\begingroup$ What about the advice given in mathoverflow.net/questions/141024/…? $\endgroup$
    – Friedrich Knop
    Commented Jun 20, 2016 at 16:16

1 Answer 1

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Consider the Lie algebra $\frak{sp}(n)$ of the symplectic group ${\rm Sp}(n)={\rm Sp}(n; \mathbb{C})\cap{\rm SO}(4n)$.

For $n=1$, let us denote by $V$ the representation of $\frak{sp}(1)$ on $\mathbb{C}^{2}$. Then, any irreducible $\frak{sp}(1)$-representation is isomorphic to some symmetric power ${\rm Sym}^{p}V$ of $V$.

For $n>1$ let us denote by $W$ the $2n$-dimensional complex representation of $\frak{sp}(n)$. Then, the symmetric powers ${\rm Sym}^{p}W$ are also irreducible. But not the exterior powers $\Lambda^{p}W$. Let us denote the kernel of the contraction $\Lambda^{p}W\to \Lambda^{p-2}W$ induced by the invariant 2-form on $W$, by $\Lambda_{0}^{p}W$. For $p\leq n$ one can show that this submodule $\Lambda_{0}^{p}W$ is also irreducible. One can moreover define the modules ${\rm Ker}\{\Lambda^{2}({\rm Sym}^{2}W)\to {\rm Sym}^{2}W\}$ and ${\rm Ker}\{\Lambda^{2}({\Lambda}^{2}_{0}W)\to {\rm Sym}^{2}W\}$, but for $n=2$ the second $\frak{sp}(n)$-module is trivial. Some details can be found in the classical book of Simon Salamon, ''Riemannian geometry and holonomy groups''. In particular, for ${\rm Sp}(2)$ see pages 80-84. Finally notice that since $\frak{sp}(2)\cong\frak{so}(5)$, the spin representation of ${\rm SO}(5)$ is isomorphic to the standard representation $W$ of ${\rm Sp}(2)$ and the standard representation of ${\rm SO}(5)$ is isomorphic with $\Lambda_{0}^{2}W$. Of course, using the LiE program you can take a quick view of the complex irreducible representations of ${\rm Sp}(2)$, in terms of highest weights.

added The irreducible representation of ${\rm Sp}(2)$ with highest weight (2, 2) is of dimension 81. Since ${\rm Sp}(2)$ is connected and simply-connected, it coincides wth ${\rm Spin}(5)$. Now, this irreducible representation is identified with the isotropy representation of the homogeneous space ${\rm SO}(14)/{\rm SO}(5)$, which is strongly isotropy irreducible. Notice finally that also the coset ${\rm Spin}(10)/{\rm Sp}(2)$ is a strongly isotropy irreducible space (of dimension 35). Here, the isotropy representation has highest weight (2, 1).

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  • $\begingroup$ Thanks for the reply. I know this much already but wonder say what the irreducible representation of highest weight $(2,2)$ is (this should come from Young diagram $(4,2)$). $\endgroup$
    – fretty
    Commented Jul 5, 2016 at 22:38
  • $\begingroup$ Thanks for the extra info. I get that this quotient has dimension 81 and that we act on it by the adjoint action but I just have two more questions. What is the explicit isomorphism $\text{Sp}(2) \rightarrow \text{SO}(5)$ and how are we viewing this as a subgroup of $\text{SO}(14)$? $\endgroup$
    – fretty
    Commented Jul 6, 2016 at 8:14
  • $\begingroup$ Alternatively what model am I going to need to use for the Lie algebra? $\endgroup$
    – fretty
    Commented Jul 6, 2016 at 8:16
  • $\begingroup$ Yes, I have seen this but have no idea how it helps me to construct an explicit isomorphism...also doesn't the isotropy rep need the Lie algebra to act? If so this creates another issue since I have no way computationally of passing from $SO(5)$ to it's Lie Algebra (unless I am being stupid). $\endgroup$
    – fretty
    Commented Jul 6, 2016 at 8:54
  • $\begingroup$ The Lie group ${\rm Sp}(2)$ is the double covering of ${\rm SO}(5)$. In particular, $\frak{sp}(2)=\frak{so}(5)$ and and ${\rm Sp}(2)\cong{\rm Spin}(5)$, see for example Theorem 5.20, p.83 of the book: Topology of Lie Groups, I and II by Mamoru Mimura and Hirosi Toda. $\endgroup$
    – 314159.
    Commented Jul 6, 2016 at 9:17

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