Proof by characters of the orthogonal-orthogonal duality and relations of Casimir invariants

TL;DR

This paper proves the orthogonal-orthogonal duality theorem using character methods, providing new derivations of Casimir invariant relations and exploring potential applications in physical systems beyond existing literature.

Contribution

It introduces a novel character-based proof of the duality theorem and derives linear relations between Casimir invariants using Young diagram geometry.

Findings

New proof of orthogonal-orthogonal duality using characters

Derived relations between Casimir invariants

Potential applications in various physical systems

Abstract

The theorem of orthogonal-orthogonal duality of Rowe, Repka, and Carvalho is proven by a method based on characters that is very different from theirs and akin to Helmers's half a century earlier proof of the analogous sympletic-symplectic duality. I demonstrate how three duality theorems listed by Rowe, Repka, and Carvalho allow very brief derivations of linear relations between the Casimir invariants of the connected representations based on the geometry of their Young diagrams, and discuss for which physical systems other than such already considered in the literature an analysis in terms of the orthogonal-orthogonal duality might be useful.

Equations

$$1,a_{p\tau }^{†},a_{p\tau }^{†}{a}_{q\upsilon }^{†},\dots .$$

$$\sum _r(⟨p|x|r⟩⟨rq|g⟩+⟨q|x|r⟩⟨pr|g⟩)=0.$$

$$⟨pq|g⟩={\delta }_{pq}$$

$$⟨\alpha lm,{\alpha }^{\prime }{l}^{\prime }{m}^{\prime }|g⟩=(-{)}^{l+m}{\delta }_{\alpha {\alpha }^{\prime }}{\delta }_{l{l}^{\prime }}{\delta }_{m,-m^{\prime }}.$$

$$x_{pq}|q⟩=|p⟩,x_{pq}|p⟩=-|q⟩,x_{pq}|r⟩=0,x_{pp}=0,p,q,r\text{ different}.$$

$$[h_p,x_{\rho }]={\rho }_p{x}_{\rho },$$

$$(h_p,x_{\rho })=0,(x_{\rho },x_{\sigma })=0,\sigma \ne -\rho ,[x_{\rho },x_{-\rho }]=(x_{\rho },x_{-\rho })\sum _p{\rho }_p{h}_p.$$

$$(x_{pq},x_{rs})={\delta }_{pq,sr}$$

$$|\pm \nu ⟩=\sqrt{\frac{1}{2}}(|p⟩\pm i|-p⟩),\nu =p>0,|\nu ⟩=|p⟩,\nu =p=0.$$

$$h_{\nu }|\pm \nu ⟩=\pm |\nu ⟩,h_{\nu }|\pi ⟩=0,\pi \ne \pm \nu .$$

$$X=\sum _{pq\tau }⟨p|x|q⟩a_{p\tau }^{†}{a}_{q\tau }.$$

$$n=\sum _{p\tau }{a}_{p\tau }^{†}{a}_{p\tau }$$

$$C_{\text{o(d)}}=\sum _{p>q}{X}_{pq}{X}_{qp}=\frac{1}{2}\sum _{pq}{X}_{pq}{X}_{qp}.$$

$$Y=\frac{1}{2}\sum _{pq\tau \upsilon }⟨p\tau |y|q\upsilon ⟩[a_{p\tau }^{†},a_{q\upsilon }]=\sum _{pq\tau \upsilon }⟨p\tau |y|q\upsilon ⟩(a_{p\tau }^{†}{a}_{q\upsilon }-\frac{1}{2}{\delta }_{p\tau ,q\upsilon }),$$

$$Y_{\tau ,-\upsilon }:=-Y_{-\upsilon ,\tau }=\sum _p{a}_{p\tau }^{†}{a}_{p\upsilon }-\frac{d}{2}{\delta }_{\tau \upsilon }.$$

$$Y=\sum _{pq\tau \upsilon }⟨p\tau ,q\upsilon |y⟩a_{p\tau }^{†}{a}_{q\upsilon }^{†},$$

$$\sum _r(⟨p|x|r⟩⟨r\tau ,q\upsilon |y⟩+⟨q|x|r⟩⟨p\tau ,r\upsilon |y⟩)=0.$$

$$Y_{\tau \upsilon }=\sum _p{a}_{p\tau }^{†}{a}_{p\upsilon }^{†}.$$

$$Y_{\alpha \beta }=\frac{1}{2}\sum _p[a_{p\alpha }^{†},a_{p\beta }^{†}],$$

$$[Y_{\alpha \beta },Y_{\gamma \delta }]={\delta }_{\beta ,-\gamma }{Y}_{\alpha \delta }-{\delta }_{\beta ,-\delta }{Y}_{\alpha \gamma }-{\delta }_{\alpha ,-\gamma }{Y}_{\beta \delta }+{\delta }_{\alpha ,-\delta }{Y}_{\beta \gamma }.$$

$$(Y_{\alpha \beta },Y_{\gamma \delta })={\delta }_{\alpha \beta ;-\delta ,-\gamma }.$$

$$C_{\text{o(2k)}}=\sum _{\alpha >\beta }{Y}_{\alpha \beta }{Y}_{-\beta ,-\alpha }=\frac{1}{2}\sum _{\alpha \beta }{Y}_{\alpha \beta }{Y}_{-\beta ,-\alpha }.$$

$$Y_{\tau \upsilon }=\sum _{pq}{a}_{p\tau }^{†}{a}_{q\upsilon }^{†}⟨pq|g⟩.$$

$$0\le {\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_{\Omega }.$$

$$l_p={\lambda }_p+p-1$$

$$l_p={\lambda }_p+p-\frac{1}{2}$$

$$\Phi =⨁{\text{X}}_{[\lambda ]}\otimes {\Psi }_{[\mu ]},$$

$$0,1,...,\Omega +k-1$$

$$\frac{1}{2},\frac{3}{2},...,\Omega +k-\frac{1}{2}$$

$$H=\sum _{\nu >0}{\varphi }_{\nu }{H}_{\nu }=\sum _{\nu >0,\tau }{\varphi }_{\nu }(a_{\nu \tau }^{†}{a}_{\nu \tau }-a_{-\nu ,\tau }^{†}{a}_{-\nu ,\tau }).$$

Full text

I Introduction

In 1961 Helmers proved a remarkable theorem that establishes a 1–1 correspondence between the irreps of two commuting symplectic Lie algebras of operators on the Fock space of several kinds of fermions inhabiting a common 1-kind configuration space of even dimension. Helmers (1961) Helmers’s article appeared in a context of contemporary efforts Bohr, Mottelson, and Pines (1958); Bogolyubov (1958); Solov’yov (1958a); *ref:Sol58b to adapt to nuclei the theory of superconductivity of Bardeen, Cooper, and Schrieffer. Bardeen, Cooper, and Schrieffer (1957a); *ref:Bar57b In the simplest application of his theorem, only one kind of nucleon, neutron or proton, is considered. A basis for the 1-nucleon configuration space may consist of one or several shells of orbits, where the orbits within a shell share the value of the angular momentum quantum number $j$, or it may be composed of pairs of time reversed stationary states in a deformed potential well. One symplectic Lie algebra is then induced by infinitesimal symplectic transformations of that space. Due to the antisymmetry of many-fermion wave functions, only irreps associated with 1-column Young diagrams occur. They are described by the quantum number of seniority defined for atoms by Racah Racah (1943) in 1943, and their carrier spaces are eigenspaces of a particular “pairing” interaction which acts only in those pairs of time reversed orbits whence Cooper pairs are built.

In this simplest case, the second symplectic Lie algebra was introduced by Kerman in the same context of adaption of the Bardeen-Cooper-Schrieffer theory to nuclei. Kerman (1961) It includes operators that create or annihilate pairs of nucleons. It was not recognized at its conception as a symplectic Lie algebra because its structure is that of $\mathfrak{sp}(2)$, which is isomorphic to $\mathfrak{sl}(2)$. Accordingly, Kerman’s Lie algebra is known to nuclear physicists as the quasi-spin algebra. The 1–1 correspondence between seniority and quasi-spin established by Helmers is at the core of an extensive analysis of the nuclear shell model pursued most prominently by Talmi and his coworkers. Talmi (1993) The virtue of the correspondence of seniority and quasi-spin at the base of this analysis lies in the fact that the quasi-spin algebra connects states of equal seniority in nuclei with different numbers of nucleons. This allows conclusions about states of a given seniority of more complex nuclei to be drawn by recourse to systems with fewer nucleons.

A note on notation: In the present article, $\mathfrak{sl}(d)$ is $A_{\Omega}$ in Cartan’s classification, Cartan (1894); Racah ; Jacobsen (1962) with $d=\Omega+1$, and $\mathfrak{gl}(d)$ is $\mathfrak{sl}(d)$ extended by a commuting 1-dimensional Lie algebra. The Lie algebra $\mathfrak{o}(d)$ is $B_{\Omega}$ or $D_{\Omega}$ with $d=2\Omega+1$ and $d=2\Omega$, respectively, and $\mathfrak{sp}(d)$ is $C_{\Omega}$ with $d=2\Omega$. Having quantum mechanical applications in mind, I understand that the base field is the field of complex numbers.

Already in 1952, Flowers generalized Racah’s treatment of systems with one kind of fermions to systems of neutrons and protons sharing a configuration space. Flowers (1952a) The states of such a system may carry irreps of the Lie algebra of infinitesimal symplectic transformations of the 1-kind configuration space associated with 2-column Young diagrams. Let $d$ be the dimension of the 1-kind configuration space. To describe $\mathfrak{sp}(d)$ irreps associated with Young diagrams with maximally 2 columns, Flowers introduces a second quantum number besides seniority, the quantum number of reduced isospin. By Helmers’s theorem, each pair of seniority and reduced isospin then corresponds 1–1 to an irrep of a number non-conserving $\mathfrak{sp}(4)$ algebra which commutes with the $\mathfrak{sp}(d)$ algebra. This work by Flowers was preceded by many studies where spatial states of the nuclear system were classified by representations of $\mathfrak{o}(d)$ associated with Young diagrams with maximally 4 columns, corresponding to the 4 dimensions of the space of spin and isospin of a nucleon. Here, $d$ denotes the dimension of the 1-nucleon spatial configuration space. See Refs. Jahn, 1950; Flowers, 1952b and refs. therein.

Independently of Helmers, Flowers and Szpikowski introduced a few years after his work a “generalized quasi-spin” algebra, which they describe as $\mathfrak{o}(5)$, and which is, in fact, identical to Helmers’s $\mathfrak{sp}(4)$ algebra. Flowers and Szpikowski (1964a) (The isomophism of these two Lie algebras is well known. Jacobsen (1962); Racah ) Shortly thereafter, and returning to a separation of the degrees of freedom of a single nucleon into spatial ones on the one hand and spin and isospin on the other, these authors proposed an $\mathfrak{o}(8)$ algebra of “quasi-spin in $LS$ coupling”. Flowers and Szpikowski (1964b) Flowers and Szpikowski calculated the spectra of pairing interactions in terms of Casimir invariants of these semi-simple Lie algebras and found that the resulting expressions were identical, upon a suitable association of quantum numbers, to such obtained in previous analyses in terms of representations of $\mathfrak{sp}(d)$ or $\mathfrak{o}(d)$. Edmonds and Flowers (1952); Bayman (1960) In the case of $\mathfrak{o}(5)$ and $\mathfrak{sp}(d)$, the association of quantum numbers could have been derived from Helmers’s theorem, had the authors been aware or it at the time of writing. Both algebras $\mathfrak{o}(5)$ and $\mathfrak{o}(8)$ have attracted much attention from the nuclear physics community within the last two decades related to discussions of the possibility of condensation of Cooper pairs built from pairs of a neutron and a proton besides an established presence of condensates of Cooper pairs built from pairs of nucleons of the same kind. Engel, Langanke, and Vogel (1996); Engel et al. (1997); Kota and Castilho Alcarás (2006); Dukelsky et al. (2006); Lerma H. et al. (2007); Rowe and Carvalho (2007); Drumev and Georgieva (2013, 2014); Márquez Romero, Dobaczewski, and Pastore (2018)

Helmers anticipates in Ref. Helmers, 1961 that results similar to his hold in $LS$ coupling, but half a century would pass before a theorem analogous to his and pertaining to the separations of fermionic degrees of freedom into such of motion in space and additional quantum numbers would appear in the physics literature. Rowe, Repka, and Carvalho (2011) (A proof pertaining to the case of $\mathfrak{o}(8)$ was published a few years earlier. Rowe and Carvalho (2007)) In Ref. Rowe, Repka, and Carvalho, 2011, Rowe, Repka, and Carvalho both present a new proof of Helmers’s theorem and prove an analogous theorem that involves orthogonal Lie algebras. I follow their terminology and name the correspondences established by these theorems the symplectic-symplectic and orthogonal-orthogonal dualities, respectively, abbreviated $\mathfrak{sp}$-$\mathfrak{sp}$ and $\mathfrak{o}$-$\mathfrak{o}$. In both cases, one of the Lie algebras generalizes the algebras $\mathfrak{sp}(d)$ and $\mathfrak{o}(d)$ mentioned above. The other one is $\mathfrak{sp}(2k)$ or $\mathfrak{o}(2k)$, where $k$ is the number of “kinds” of particles. To make the latter concept concrete, in the $\mathfrak{sp}$-$\mathfrak{sp}$ duality, $k$ equals 1 for systems of only electrons, only neutrons or only protons, and $k=2$ for the system of neutrons and protons. In relation to the $\mathfrak{o}(8)$ algebra of Flowers and Szpikowski, $k$ equals 4 corresponding to the 4 linearly independent states of spin and isospin of a nucleon. The algebra $\mathfrak{sp}(d)$ or $\mathfrak{o}(d)$ will be referred to as the number conserving Lie algebra and the algebra $\mathfrak{sp}(2k)$ or $\mathfrak{o}(2k)$ as the number non-conserving Lie algebra.

It should be specified at this point that duality in the sense of Ref. Rowe, Repka, and Carvalho, 2011 and the present article requires more than commuting Lie algebras and a 1–1 correspondence of representations. Each product of connected representations must also be realized with multiplicity 1 on the total Hilbert space, which is the Fock space in this case.

The proofs in Ref. Rowe, Repka, and Carvalho, 2011 take recourse to yet another duality, which is proven there, as well. The authors call it a unitary-unitary duality, but the name of a $\mathfrak{gl}$-$\mathfrak{gl}$ duality would be equally justified. It refers to an observation that is known and applied since long ago in the theory of many-fermion systems: Let the configuration space of one fermion be spanned by wave functions $\chi(p)\psi(\tau)$, where $p$ and $\tau$ are sets of quantum numbers such as spatial and spin quantum numbers. Let $\mathfrak{gl}(d)$, where $d$ is the number of values of $p$, denote the tensor representation on the space of functions $\phi(p_{1},\dots,p_{n},\tau_{1},\dots,\tau_{n})$ of the algebra of infinitesimal linear transformations of the function $\chi$, and $\mathfrak{gl}(k)$ similarly in terms of $\psi$ and the number $k$ of values of $\tau$. Then $\mathfrak{gl}(d)$ and $\mathfrak{gl}(k)$ are dual on the space of $n$-fermion wave functions in the sense above, and the connected irreps are associated with Young diagrams which result from one another by the interchange of rows and columns. Such Young diagrams are called conjugate. The assertion follows from the facts that $\phi$ can be expanded on products $\chi(p_{1},\dots,p_{n})\psi(\tau_{1},\dots,\tau_{n})$, and that, say, $\mathfrak{gl}(d)$ and the symmetric group $S_{n}$ are dual on the space of functions $\chi$. Weyl (1939); Littlewood (1940) (The Schur-Weyl duality in the terminology of Ref. Rowe, Repka, and Carvalho, 2011.) Antisymmetrization in the pairs $(p_{i},\tau_{i})$ leaves one irreducible $\mathfrak{gl}(d)\oplus\mathfrak{gl}(k)$ module for each pair of conjugate Young diagrams such that the corresponding $S_{n}$ irreps can be realized on functions $\chi$ and $\psi$.

In Ref. Rowe, Repka, and Carvalho, 2011 the proofs of the $\mathfrak{sp}$-$\mathfrak{sp}$ and $\mathfrak{o}$-$\mathfrak{o}$ dualities proceed from the observation that the number conserving Lie algebra is contained in a $\mathfrak{gl}(d)$ algebra and the number non-conserving one contains a $\mathfrak{gl}(k)$ algebra. Each irreducible module of their direct sum (more precisely in the $\mathfrak{o}$-$\mathfrak{o}$ case, of the outer product $O(d)\otimes SO(2k)$ of the full and proper orthogonal groups) is then shown to contain a state which has the highest weight with respect to irreps of both $\mathfrak{gl}$ algebras. Helmers’s original proof of the $\mathfrak{sp}$-$\mathfrak{sp}$ duality is very different from this and more direct. It uses the expression for the $\mathfrak{sp}$ characters found early in the last century. Weyl (1939); Littlewood (1940) It seems instructive to prove also the $\mathfrak{o}$-$\mathfrak{o}$ duality in this manner. This is done in Secs. II–IV of the present article, where the theorem is also stated more symmetrically in $\mathfrak{o}(d)$ and $\mathfrak{o}(2k)$ than in Ref. Rowe, Repka, and Carvalho, 2011.

In the applications to nuclear spectroscopy mentioned above, Casimir invariants play a central role. The Casimir invariants of dually connected irreps must be related, and these relations turn out to be linear. Some of these relations were derived algebraically, and such derivations were seen occasionally as proofs of the duality, which of course they are not. Helmers (1961); Kota and Castilho Alcarás (2006); Drumev and Georgieva (2013, 2014) It amounts to essentially the same that in several cases the spectra of pairing interactions where expressed by Casimir invariants of either one of a pair of dual Lie algebras and the expressions obtained were found to agree upon suitable associations of quantum numbers. Talmi (1993); Flowers and Szpikowski (1964a, b) (In the particular case of the $\mathfrak{sp}(d)$–$\mathfrak{sp}(2)$ duality, Talmi (1993) the 1–1 correspondence actually follows from this comparison, but multiplicity 1 of the products of representations does not.) In Secs. V–VI, I show how to derive in a very simple manner based on the geometry of the Young diagrams the linear relations of Casimir invariants directly from the associations of representation according to the $\mathfrak{o}$-$\mathfrak{o}$, $\mathfrak{sp}$-$\mathfrak{sp}$, and $\mathfrak{gl}$-$\mathfrak{gl}$ duality theorems. A more complicated, algebraic derivation akin to those of Refs. Helmers, 1961 and Kota and Castilho Alcarás, 2006 is given in one case for comparison.

Before summarizing the article in Sec. VIII, I address in Sec. VII the question of possible applications of the orthogonal-orthogonal duality to the analysis of actual physical systems other than such already considered in the literature.

II Orthogonal Lie algebras

To prepare the proof of the $\mathfrak{o}$-$\mathfrak{o}$ duality in the manner of Helmers, I describe in this section the construction of the two commuting $\mathfrak{o}$ algebras. A number $k$ of kinds of fermions is considered, and the kind is denoted by letters $\tau,\upsilon,\dots$ . These fermions inhabit a common 1-kind configuration space of dimension $d$ with orthonormal basic states denoted by $|p\rangle,|q\rangle,\dots$ . The annihilator of a fermion of kind $\tau$ in the state $|p\rangle$ is denoted by $a_{p\tau}$. By definition, the Fock space $\Phi$ is spanned by the states generated from the vacuum by the operators

[TABLE]

II.1 Number conserving Lie algebra

A Lie algebra of generators $x$ of orthogonal transformations of the 1-kind configuration space is defined by the conservation of a non-singular symmetric tensor $\langle pq|g\rangle=\langle qp|g\rangle$ in the sense that

[TABLE]

It will be assumed that

[TABLE]

for some choice of the states $|p\rangle$. The matrix $\langle p|x|q\rangle$ then is antisymmetric. If the 1-kind configuration space carries integral angular momentum quantum numbers $l$, one can write $|p\rangle=|\alpha lm\rangle$, where $m$ is the magnetic quantum number. In atomic and nuclear physics, $g$ is then often taken to be the Wigner metric Edmonds (1957)

[TABLE]

Because this can be given the form (3) by a unitary transformation, it is encompassed by the assumption above. Using the metric (4) instead of (3) only adds complication.

Operators $x_{pq}$ can be defined by

[TABLE]

They satisfy $x_{pq}=-x_{qp}$, and those with $p>q$ form a basis for the Lie algebra. The index $p$ of the basic 1-kind states states $|p\rangle$ may be assumed to take the values $-\Omega,-\Omega+1,$ $\ldots,\Omega$ with 0 omitted when $d$ is even. The operators $h_{p}=-ix_{p,-p},\ p>0,$ then form a basis for a Cartan subalgebra which gives the roots of Ref. Racah, . Because the root diagram is invariant under reflections and permutations of the coordinate axes, the Killing form $(x,x^{\prime})$ can be so renormalized that $(h_{p},h_{q})=\delta_{pq}$. I here include for convenience the case $d=2$ when the Killing form vanishes because the Lie algebra is 1-dimensional and therefore Abelian. For $d=1$ the Lie algebra is 0-dimensional (has one element, 0) so the Killing form is undefined.

For every root $\rho$ an operator $x_{\rho}$ is determined within a normalization by

[TABLE]

and the operators $h_{p}$ and $x_{\rho}$ form a basis for the Lie algebra. The entire Killing form is then determined by the relations Jacobsen (1962); Racah

[TABLE]

Transformation to the basis of operators $x_{pq}$ gives the simple result

[TABLE]

The basic operators $h_{p}$ of the Cartan subalgebra are diagonalized by the unitary transformation to basic states $|\nu\rangle,|\pi\rangle,\dots$ defined by Weyl (1939)

[TABLE]

In fact,

[TABLE]

The Lie algebra of operators $x$ is faithfully represented on $\Phi$ by the operators

[TABLE]

The representative of $h_{p}$ will be denoted by $H_{p}$. Evidently every $X$ commutes with the total number

[TABLE]

of fermions. From now on, the symbol $\mathfrak{o}(d)$ is reserved for the orthogonal Lie algebra of these operators $X$. By Eq. (8), its Casimir operator C_{\text{\mathfrak{o}(d)}} is given by

[TABLE]

II.2 Number non-conserving Lie algebra

I turn to the construction of the number non-conserving orthononal Lie algebra. To this end let $\phi,\chi,\dots$ denote arbitrary linear combinations of the fermion fields $a_{p\tau}$ and $a^{\dagger}_{p\tau}$, and let $Y$ be any linear combination of commutators $[\phi,\chi]$. The set of such $Y$ is closed under commutation.

One may now inquire which $Y$ commute with every $X$. Considering first operators

[TABLE]

one finds that each matrix in $p$ and $q$ with elements $\langle p\tau|y|q\upsilon\rangle$ must commute with every matrix $\langle p|x|q\rangle$. For $d\neq 2$ the representation of the Lie algebra of operators $x$ by itself is irreducible. (Cf. Sec. III.1. For $d>1$ this representation is described by the 1-cell Young diagram. For $d=1$ the 1-kind space is trivially irreducible.) By Schur’s lemma, each matrix $\langle p\tau|y|q\upsilon\rangle$ is then proportional to the unit matrix, so the space of operators $Y$ of the form (14) that commute with every $X$ is spanned by the operators

[TABLE]

As to operators

[TABLE]

the requirement is

[TABLE]

By the antisymmetry of the matrix $\langle p|x|y\rangle$ the matrix in $p$ and $q$ with elements $\langle p\tau,q\upsilon|y\rangle$ then commutes with $\langle p|x|q\rangle$. For $d\neq 2$ it follows again that $\langle p\tau,q\upsilon|y\rangle$ is proportional to $\delta_{pq}$. The space of operators $Y$ of the form (16) that commute with every $X$ is then spanned by the operators

[TABLE]

The hermitian conjugates of the operators (18) will be denoted by $Y_{-\upsilon,-\tau}$ and span the space of linear combinations of products of pairs of annihilators that commute with every $X$.

All the basic operators defined above have the form

[TABLE]

where $\alpha$ and $\beta$ take values $\pm 1,\pm 2,\dots,\pm k$, and $a_{p,-\tau}:=a^{\dagger}_{p\tau}$. One easily derives

[TABLE]

By comparison with Ref. Racah, it follows that the span of the set of operators (19) is an $\mathfrak{o}(2k)$ algebra, and the symbol $\mathfrak{o}(2k)$ is reserved from now on for this orthogonal Lie algebra. The operators $Y_{\tau,-\tau}$ form a basis for a Cartan subalgebra which gives the roots of Ref. Racah, . Renormalizing the Killing form $(Y,Y^{\prime})$ so that these operators have squared Killing norm 1 results in

[TABLE]

The Casimir operator C_{\text{\mathfrak{o}(2k)}} is then given by

[TABLE]

For $d=2$ the algebra $\mathfrak{o}(2k)$ is not the maximal set of operators $Y$ that commute with every $X$. The following discussion holds, anyway, in this case, as well. (For $d=2$ the maximal set forms a $\mathfrak{gl}(2k)$ algebra and contains $\mathfrak{o}(d)$. If also $k>1$, the Lie algebra $\mathfrak{o}(d)$ is the maximal subalgebra of this $\mathfrak{gl}(2k)$ algebra that commutes with $\mathfrak{o}(2k)$. If $k=1$, both Lie algebras are 1-dimensional, and they span a 2-dimensional Abelian Lie algebra.)

The unitary transformation that transforms the metric (3) to a general symmetric $g$ transforms the operators (18) to

[TABLE]

The form of the operators  (15) does not change by this transformation.

III Orthogonal-orthogonal duality

III.1 Representations of orthogonal Lie algebras

Before turning to the proof of the duality theorem it is necessary to recapitulate some facts of representations of orthogonal Lie algebras. My notation refers to $\mathfrak{o}(d)$, but the discussion applies analogously to $\mathfrak{o}(2k)$ upon evident substitutions. I shall similarly not in the subsequent sections state explicitly results for $\mathfrak{o}(2k)$, $\mathfrak{sp}(2k)$, or $\mathfrak{gl}(k)$ which follow by analogy from those for $\mathfrak{o}(d)$, $\mathfrak{sp}(d)$, or $\mathfrak{gl}(d)$.

The Lie algebra $\mathfrak{o}(d)$ has representations described by partitions $[\lambda]$, which are sequences of parts $\lambda_{1},\lambda_{2},\dots,\lambda_{\Omega}$ with

[TABLE]

(See Refs. Weyl, 1939; Littlewood, 1940, noticing that the Lie algebra generates the proper orthogonal group.) Unconventionally, I number the parts in non-decreasing order, which will turn out convenient. The convention is to number them with the largest first. Either all $\lambda_{p}$ are integral or all of them are half-integral. Representations with half-integral parts are spin representations. The representation described by the partition $[\lambda]$ is irreducible except when $d$ is even and $\lambda_{1}>0$. When this happens, the representation splits into two inequivalent irreps. They may be distinguished by opposite signs of $\lambda_{1}$, in which case $|\lambda_{1}|$ replaces $\lambda_{1}$ in the inequalities (24). Then $\lambda_{p}$ is always the eigenvalue of $H_{p}$ on the vector of highest weight of an irreducible module.Börner (1955) I consider here only non-negative $\lambda_{1}$ and understand that $[\lambda]$ then describes the total, reducible representation when $d$ is even and $\lambda_{1}>0$. By definition, the empty partition describes a 1-dimensional representation by the constant 0.

A partition is visualized by a Young diagram, whose rows have lengths $\lambda_{p}$ and are stacked from bottom to top in the order of $p$. In particular the Young diagram of a spin representation thus has a leftmost column of width $\frac{1}{2}$ and height $\Omega$.

In the theory of characters a central role is played by the numbers

[TABLE]

when $d$ is even, and

[TABLE]

when $d$ is odd. The number $2(l_{p}-\lambda_{p})$ is the $p$th component of the sum of positive roots. I call the strictly increasing sequence of the numbers $l_{p}$ associated with a given partition $[\lambda]$ the unfolded partition and denote it by $[[l]]$.

III.2 Theorem

The theorem of $\mathfrak{o}$-$\mathfrak{o}$ duality can now be stated as follows.

Theorem: The Fock space $\Phi$ has the decomposition

[TABLE]

where $\textup{X}_{[\lambda]}$ and $\Psi_{[\mu]}$ carry single representations of $\mathfrak{o}(d)$ and $\mathfrak{o}(2k)$, respectively, described by the partitions $[\lambda]$ and $[\mu]$. In the sum, each pair of $[\lambda]$ and $[\mu]$ that satisfies the following criterion appears exactly once. Let $[[l]]$ and $[[m]]$ be the corresponding unfolded partitions, and consider the set of numbers that appear in either $[[l]]$ or $[[m]]$. The criterion is that this set consists of the numbers

[TABLE]

when $d$ is even, and

[TABLE]

when $d$ is odd.

It may be noticed that when $d$ is odd, so as when the 1-kind space is a single $l$ shell, the $\mathfrak{o}(2k)$ representations are spin representations. The $\mathfrak{o}(d)$ representations are always non-spin representations.

The connection between the partitions $[\lambda]$ and $[\mu]$ is analogous to that of the Helmers theorem for symplectic Lie algebras, Helmers (1961) and can, like the latter, be expressed in geometric terms: Consider a rectangle of width $k$ and height $d/2$. In this rectangle, place the $[\lambda]$ Young diagram at the upper left corner and the $[\mu]$ Young diagram at the lower right corner, rotated 180∘ and reflected in the bisector of its right angle. Then the two diagrams must fill the rectangle without overlap. See Fig. 1 of Ref. Helmers, 1961 for an illustration. It follows that $[\lambda]$ has no part greater than $k$, and $[\mu]$ has no part greater than $d/2$. Proving that this criterion is equivalent to that of the theorem goes as in the symplectic case with a very minor modification due to a different relation corresponding to the relations (25) and (26) and a different sequence corresponding to the sequences (28) and (29), cf. Sec. V.1. I leave it to any interested reader to work it out based on Ref. Helmers, 1961.

Yet another formulation is that for each $p$ the $p$th row of the $[\lambda]$ diagram with the numbering above and the $p$th column from the left of the $[\mu]$ diagram, not counting a possible half-width column, have total length $k$, or, equivalently, that for each $\tau$ the $\tau$th row of the $[\mu]$ diagram with the analogous numbering and the $\tau$th column from the left of the $[\lambda]$ diagram have total length $d/2$.

IV Proof of the theorem

By Eq. (10) the general member of the $\mathfrak{o}(d)$ Cartan subalgebra can be written

[TABLE]

The trace

[TABLE]

over the carrier space of the representation $[\lambda]$ of

[TABLE]

where

[TABLE]

is a polynomial in $\epsilon_{\nu}^{\;\frac{1}{2}}$ which specifies the representation uniquely. It is known as the character of the representation. Jacobsen (1962) Similarly the trace

[TABLE]

over the carrier space of the representation $[\mu]$, where

[TABLE]

with

[TABLE]

is a character of that representation.

Because $\mathfrak{o}(d)$ and $\mathfrak{o}(2k)$ commute, the direct sum of their Cartan subalgebras is a Cartan subalgebra of their direct sum, so the character $\chi$ of a representation of \text{\mathfrak{o}(d)}\oplus\text{\mathfrak{o}(2k)} is given by

[TABLE]

It is a polynomial in $\epsilon_{\nu}^{\;\frac{1}{2}}$ and $\eta_{\tau}^{\;\frac{1}{2}}$. When $\chi$ is evaluated on $\Phi$, proving the theorem requires the verification of the identity

[TABLE]

where the sum runs over the same combinations of $[\lambda]$ and $[\mu]$ as in Eq. (27).

When $d$ is even and $\lambda_{1}>0$, the character $\chi_{[\lambda]}$ given by the expressions (49) and (56) below is the sum of characters of the two inequivalent irreps which compose the representation $[\lambda]$. Littlewood (1940) The proof then asserts that both these irreps are present together. Similarly when $\mu_{1}>0$.

The verification of the identity (38) proceeds separately for even and odd $d$. Before entering these separate cases, I make a definition. The symbol

[TABLE]

where $f_{1},\dots,f_{n}$ are any functions, denotes the determinant whose rows have the form shown for $n$ different values of $z$ specified in the context

IV.1 Even $d$

The verification of the identity (38) in the case of even $d$ follows closely Helmer’s proof of the $\mathfrak{sp}$-$\mathfrak{sp}$ duality. Helmers (1961) It is noticed that $\Phi$ is the direct product of 2-dimensional spaces, one for each pair of $\nu$ and $\tau$. Basic states of each 2-dimensional space correspond to the state $|\nu\tau\rangle$ of 1 fermion being empty or occupied. Each such space contributes to $\chi$ a factor

[TABLE]

The pair of states $|\pm\nu,\tau\rangle$, where $\nu>0$, then gives the factor

[TABLE]

where

[TABLE]

One arrives at

[TABLE]

with

[TABLE]

Evidently $F(\epsilon)=0$ for $\epsilon=-\eta_{\tau}$. Using

[TABLE]

one gets

[TABLE]

Here, in the first determinant, $\epsilon$ takes the values $\epsilon_{\nu}$, and in the second one, $\eta$ takes the values $\eta_{\tau}$. In the third determinant, $\epsilon$ takes the values $-\eta_{\tau}$ in the first $k$ rows, and $\epsilon_{\nu}$ in the remaining $\Omega$ rows.

Now, using

[TABLE]

one gets

[TABLE]

where the sum runs over all the pairs of unfolded partitions $[[l]]$ and $[[m]]$ of the theorem, and $\epsilon$ and $\eta$ take the values $\epsilon_{\nu}$ and $\eta_{\tau}$, respectively. The identity (38) now follows by Weyl (1939); Littlewood (1940)

[TABLE]

and the analogous expression for $\chi_{[\mu]}$ (written explicitly in Eq. (56)).

IV.2 Odd $d$

The idea of the verification of the identity (38) for odd $d$ is the same, but the details are slightly more involved. There is now a state $|\nu\rangle$ with $\nu=0$, which contributes to $\chi$ an extra factor

[TABLE]

This factor can be combined with $|c_{0}(\eta),c_{1}(\eta),\dots,c_{k-1}(\eta)|$ to give

[TABLE]

where

[TABLE]

and $(-z)^{\alpha}$ is shorthand for $\exp(i\alpha\pi)z^{\alpha}$. One gets

[TABLE]

with the same $\epsilon$ and $\eta$ as in Eq. (46).

Using

[TABLE]

for half-integral $\alpha$ and $\beta$, one obtains

[TABLE]

By Weyl (1939); Littlewood (1940)

[TABLE]

the identity (38) follows.

V Symplectic and general linear Lie algebras

V.1 Symplectic Lie algebras

The algebras $\mathfrak{sp}(d)$ and $\mathfrak{sp}(2k)$ are very similar to $\mathfrak{o}(d)$ and $\mathfrak{o}(2k)$. A Lie algebra of generators $x$ of symplectic transformations of the 1-kind configuration space is defined by the conservation of a non-singular antisymmetric tensor $\langle pq|g\rangle=-\langle qp|g\rangle$ in the sense of Eq. (2). The non-singularity of $g$ requires that $d$ is even. In terms of these $x$ the members $X$ of $\mathfrak{sp}(d)$ are then defined by Eq. (11). The index $p$ of basic, orthonormal 1-kind states $|p\rangle$ may take the values $-\Omega,-\Omega+1,\dots,\Omega$ with 0 omitted. It will be assumed that these states can be so chosen that

[TABLE]

with $\sigma_{p}=\pm 1$ for $p\gtrless 0$. In atomic and nuclear physics, the 1-kind configuration space often carries half-integral angular momentum quantum numbers $j$, and $g$ is chosen to be the Wigner metric

[TABLE]

in a notation as in Eq. (4). This $g$ can be given the form (57) by a unitary transformation of the 1-kind configuration space.

The operators $x_{pq}$ defined by

[TABLE]

satisfy $x_{pq}=x_{qp}$. Those with $p\geq q$ form a basis for the Lie algebra. A basis for a Cartan subalgebra which gives the roots of Ref. Racah, is formed by the operators $x_{p,-p}$ with $p>0$. Renormalizing the Killing form $(x,x^{\prime})$ so that these operators have squared Killing norm 1 results in

[TABLE]

This gives the Casimir operator

[TABLE]

An analysis like that of Sec. II.2 gives the maximal Lie algebra spanned by commutators $[\phi,\chi]$ that commute with every $X$. Its elements of the form (14) are given by Eq. (15) while those of the form (16) are given by Eq. (23), which for the metric (57) evaluates to

[TABLE]

Including the Hermitian conjugates $Y_{-\tau,-\upsilon}$ of the latter renders the set maximal. With indices $\alpha,\beta,\dots$ as in Eq. (20), one may verify the commutation relations

[TABLE]

which identify the resulting Lie algebra as an $\mathfrak{sp}(2k)$ algebra. Racah The operators $Y_{\tau,-\tau}$ again form a basis for a Cartan subalgebra which gives the roots of Ref. Racah, , and renormalizing the Killing form so that these operators have squared Killing norm 1 results in an expression for the Casimir operator analogous to Eq. (61). The discussion in Sec. III applies almost verbatim to the $\mathfrak{sp}(d)$-$\mathfrak{sp}(2k)$ duality, the only differences being that only integral parts occur, every partition describes an irreducible representation, and Eqs. (25) and (26) are replaced by

[TABLE]

and the sequences (28) and (29) by

[TABLE]

V.2 General linear Lie algebras

The general linear Lie algebra $\mathfrak{gl}(d)$ is induced in the manner of Secs. II.1 and V.1 by the Lie algebra of arbitrary linear transformations $x$ of the 1-kind configuration space. The latter has a basis of operators $x_{pq}$ defined by

[TABLE]

This Lie algebra is not semi-simple, but its special linear subalgebra of traceless linear transformations is simple. The Killing form of the latter can be renormalized to be the restriction to the subalgebra of the form

[TABLE]

defined on the general linear Lie algebra. It is therefore customary in nuclear theory to define a Killing form for the general linear Lie algebra by Eq. (67), and accordingly a Casimir operator C_{\text{\mathfrak{gl}(d)}} of $\mathfrak{gl}(d)$ by Edmonds and Flowers (1952)

[TABLE]

Because its difference from the Casimir operator C_{\text{\mathfrak{sl}(d)}} of the special linear subalgebra $\mathfrak{sl}(d)$ is a term in an operator which spans the commuting 1-dimensional extension from $\mathfrak{sl}(d)$ to $\mathfrak{gl}(d)$, the operator C_{\text{\mathfrak{gl}(d)}} is a $\mathfrak{gl}(d)$ invariant.

The irreps of $\mathfrak{gl}(d)$ that are realized on $\Phi$ are described by partitions with at most $d$ non-negative, integral parts. Unlike the cases of the $\mathfrak{o}(d)$ and $\mathfrak{sp}(d)$, I obey in this case the convention, which is to number the parts in non-increasing order,

[TABLE]

The Lie algebra $\mathfrak{gl}(k)$ results from interchanging the roles of the indices $p$ and $\tau$. Specifically, Eq. (11) is replaced by

[TABLE]

where $\langle\tau|y|\upsilon\rangle$ is any $k\times k$ matrix. This renders $\mathfrak{gl}(k)$ analogous to $\mathfrak{gl}(d)$ in every respect. The theorem of $\mathfrak{gl}$-$\mathfrak{gl}$ duality was stated already in the introduction. The connected irreps have conjugate Young diagrams. Therefore the $\mathfrak{gl}(d)$ parts do not exceed $k$ and the $\mathfrak{gl}(k)$ parts do not exceed $d$.

VI Relations of Casimir invariants

VI.1 Derivations from duality

For any one of the Lie algebras $\mathfrak{g}(d)$ introduced above, the eigenvalue of the Casimir operator C_{\text{\mathfrak{g}(d)}} on the carrier space of an irrep $[\lambda]$ is denoted by C_{\text{\mathfrak{g}(d)}}([\lambda]) and referred to as the Casimir invariant of that irrep. For the semi-simple Lie algebras $\mathfrak{o}(d)$ and $\mathfrak{sp}(d)$, a general formula derived by Racah Racah (1950, ) gives

[TABLE]

For $\mathfrak{o}(d)$ and even $d$, I have here temporarily allowed either sign of $\lambda_{1}$ when it differs from 0. However, in this case the expression (71) does not depend on the sign of $\lambda_{1}$, so the Casimir invariant is common to both irreducible constituents of the representation described by the partition $[\lambda]$ defined in cf. Sec III.1, which makes it, actually, a function of this partition.

The formula (71) can be given a geometric form: Assume the usual arrangement of the $[\lambda]$ Young diagram with horizontal rows of lengths $\lambda_{p}$, left justified from bottom to top in the order of $p$, and assume that the cells are unit squares except for the half-width cells of the spin representations of $\mathfrak{o}(d)$. Then place the origin of a Cartesian coordinate system at the distance $d/2$ vertically below the upper left corner of the diagram. It is straightforward to show that Eq. (71) can then be written

[TABLE]

with the integration extended over the area of the diagram. The upper and lower signs apply to $\mathfrak{o}(d)$ and $\mathfrak{sp}(d)$, respectively.

The configuration where the $[\lambda]$ and $[\mu]$ diagrams fill a rectangle of width $k$ and height $d/2$ as described in Sec. III.2 results from first placing them so that their coordinate systems coincide and then reflecting the $[\mu]$ diagram in the line $x=y$. As the integral (72) is invariant under this reflection, one gets C_{\text{\mathfrak{g}(d)}}([\lambda])+C_{\text{\mathfrak{g}(2k)}}([\mu]) by integration over the entire rectangle. Hence, when $[\lambda]$ and $[\mu]$ are dual representations,

[TABLE]

The principle of derivation of Racah’s formula can be extended to $\mathfrak{gl}(d)$ with the result Edmonds and Flowers (1952)

[TABLE]

When the origin of the coordinate system is now placed at the upper left corner of the Young diagram and the $y$ axis is turned downwards, this becomes

[TABLE]

Using that conjugate Young diagrams are reflections of one another and the area of the diagram equals $n$, one gets for dual representations $[\lambda]$ and $[\mu]$ that

[TABLE]

Before closing this section, I notice that the integral formulas (72) and (75) allow simple derivations of formulas for C_{\text{\mathfrak{g}(d)}}([\lambda]) in terms of the column heights $\nu_{i}$ of the Young diagram, numbered by $i$ from the left, not counting the half-width column in the case of spin representations of $\mathfrak{o}(d)$. One gets

[TABLE]

VI.2 Algebraic derivation

For $k=4$, Eqs. (73) and (76) give

[TABLE]

These relations were derived previously by Kota and Castilho Alcarás Kota and Castilho Alcarás (2006) using an algebraic method. It is instructive to compare their method with that of the preceding subsection. I confine myself to deriving Eq. (73) for the case of the $\mathfrak{o}$-$\mathfrak{o}$ duality in this way. One aim of showing this derivation is to demonstrate that such algebraic work gets simpler without recourse to the Wigner metric (4) and angular momentum algebra.

I use the expressions (13) and (22) for the Casimir operators. Substitution by Eq. (11) gives

[TABLE]

By the commutations relations (20) and Eqs. (15) and (18), one gets

[TABLE]

Adding Eqs. (79) and (80) results in Eq. (73) once again.

For the $\mathfrak{sp}$-$\mathfrak{sp}$ duality, Eq. (73) was derived in this way in Ref. Helmers, 1961, and Eq. (76) results after some commutations when the term $\sum_{pq\tau\upsilon}a_{p\tau}^{\dagger}a_{p\upsilon}^{\dagger}a_{q\upsilon}a_{q\tau}$ is moved from the sum in Eq. (80) to the sum in Eq. (79). (When the metric (3) is equivalent to the Wigner metric (4), this term is a pairing interaction, and analogously in the symplectic case. Considering the sums with and without it then also gives expressions for the eigenvalue of this interaction in terms of Casimir invariants. Racah (1949); Edmonds and Flowers (1952); Racah (1952); Helmers (1961); Flowers and Szpikowski (1964a, b); Kota and Castilho Alcarás (2006); Rowe and Carvalho (2007))

VII Orthogonal-orthogonal dualities in physics

The only case of an $\mathfrak{o}$-$\mathfrak{o}$ duality which is known to me to have been discussed in relation to actual physical systems is the one where $\mathfrak{o}(2k)$ is the $\mathfrak{o}(8)$ algebra of Flowers and Szpikowski. It seems reasonable to address the question whether other cases than $k=4$ could have meaningful applications.

For $k=1$ the 1-kind configuration space contains all possible states of one fermion. One may think of an atomic or nuclear shell. The $\mathfrak{o}(d)$ representations have Young diagrams with at most 1 column, so their carrier spaces are characterized by values $n\leq d/2$ and is isomorphic to the space of all antisymmetric functions of $p_{1},\dots,p_{n}$. The carrier spaces of the $\mathfrak{o}(2k)$ representations have two basic states with $n=d/2\pm\mu_{1}$ except when $\mu_{1}=0$. The $\mathfrak{o}$-$\mathfrak{o}$ duality thus imposes identical structures on the two subspaces of $\Phi$ with these numbers of fermions. This reflects the well known particle-hole symmetry, Condon and Shortley (1935); Racah (1942); Bell (1959) which thus turns out to be encoded in the $\mathfrak{o}$-$\mathfrak{o}$ duality. For $\mu_{1}=0$, corresponding to $n=d/2$, particle-hole conjugation maps the $\mathfrak{o}(d)$ representation space onto itself, and the two irreps into which it splits in this case are characterized by opposite particle-hole conjugation parities.

For $k=2$ the states $|p\rangle$ could be states of motion in space, and $\tau=1$ and 2 could denote spin directions $\downarrow$ and $\uparrow$. This could describe an atomic shell or a spin saturated shell of neutron orbits or of proton orbits in a semi-magic nucleus. It is well known that $\mathfrak{o}(2k)=\mathfrak{o}(4)$ is not simple but splits into two commuting $\mathfrak{o}(3)$ algebras. Racah ; Jacobsen (1962) These can be chosen to be spanned by the components of the total spin

[TABLE]

where $\bm{s}$ is the spin vector acting on the span of $|\!\!\downarrow\rangle$ and $|\!\!\uparrow\rangle$, and by the components

[TABLE]

of a “spin quasi-spin” $\bm{Q}$. I denote accordingly the two $\mathfrak{o}(3)$ algebras by $\mathfrak{o}(3)_{S}$ and $\mathfrak{o}(3)_{Q}$. An irrep of the $\mathfrak{o}(4)$ algebra is described by a pair of the single parts $S$ and $Q$ of the $\mathfrak{o}(3)$ partitions (related to the Casimir invariants ${\bm{S}}^{2}=S(S+1)$ and ${\bm{Q}}^{2}=Q(Q+1)$).

The Lie algebra of operators $Y$ is isomorphic to that of transformations

[TABLE]

of the space of fermion fields $\phi$. As the former faithfully represents the $d=1$ Lie algebras, this then holds for the latter, as well. For $d=1$ and $k=2$ the space of fermion fields $\phi$ is 4-dimensional. Swapping $a_{1\downarrow}$ and $a^{\dagger}_{1\downarrow}$ in the composition of $\phi$ is a reflection of this space. In combination with the transformations $\mathcal{Y}$, which generate the proper orthogonal group $SO(4)$, it therefore generates the full orthogonal group $O(4)$. Swapping $a_{1\downarrow}$ and $a^{\dagger}_{1\downarrow}$ in the composition of $\phi$ is equivalent to doing so in the composition of the operator $Y$, which maps to swapping $a_{p\downarrow}$ and $a^{\dagger}_{p\downarrow}$ in the composition of $Y$ for every $p$ when $d$ is arbitrary. The latter operation is equivalent, in turn, to the operation $P$ of swapping emptiness and occupation of the states $|p\!\!\downarrow\rangle$ in the composition of a state in $\Phi$. Appending $P$ to $\mathfrak{o}(4)$ therefore generates a representation of $O(4)$ on $\Phi$.

Swapping $a_{p\downarrow}$ and $a^{\dagger}_{p\downarrow}$ in the expressions (81) and (82) swaps $\bm{S}$ and $\bm{Q}$. It follows that the irreps of $O(4)$ are direct sums of two $\mathfrak{o}(3)_{S}\oplus\mathfrak{o}(3)_{Q}$ irreps with swapped, different $S$ and $Q$ or a single such irrep with equal $S$ and $Q$. The weight components $w_{\tau}$ are the eigenvalues of $Y_{\tau,-\tau}$ in the representation module, which are given, by Eqs. (15), (81) and (82), by $w_{\tau}=Q_{0}\mp S_{0}$ for $\tau=1$ and 2. The highest weight of an $\mathfrak{o}(3)_{S}\oplus\mathfrak{o}(3)_{Q}$ irrep then has $w_{\tau}=Q\mp S$, so $(\mu_{1},\mu_{2})=(|S-Q|,S+Q)$. The quantum numbers $S$ and $Q$ thus determine the partition $[\mu]$ and, in turn, by the $\mathfrak{o}$-$\mathfrak{o}$ duality, the partition $[\lambda]$. Basic states with a given $n$ belong to representations of $\mathfrak{o}(d)\oplus\mathfrak{o}(3)_{S}$ described by $[\lambda]$ and $S$, and $\mathfrak{o}(3)_{Q}$ connects analogous states of this form with $d-2Q\leq n\leq d+2Q$ and even $n-(d-2Q)$. In this way the spin quasi-spin $\bm{Q}$ is analogous to Kerman’s quasispin pertaining to spin non-saturated shells. Like Flowers’s reduced isospin, the quantum number $\sigma=(\mu_{2}-\mu_{1})/2=\min(S,Q)$ could be called a reduced spin.

The seniority $v_{\text{R}}$ defined in Ref. Racah, 1943 is the first $n$ in a chain of connected states, so $v_{\text{R}}=d-2Q$. It equals the area $v=d-2\max(S,Q)$ of the $\mathfrak{o}(d)$ Young diagram only when $S\leq Q$. (It is the height of the 1-column Young diagram of the enclosing $\mathfrak{sp}(2d)$ irrep dual to the $\mathfrak{o}(3)_{Q}$ irrep. Racah (1949)) When $S\leq Q$ one has $v_{\text{R}},S=v,\sigma$, so C_{\text{\mathfrak{o}(d)}}([\lambda]) is the same function of $v_{\text{R}}$ and $S$ as of $v$ and $\sigma$. Because C_{\text{\mathfrak{o}(4)}}([\mu]), and therefore C_{\text{\mathfrak{o}(d)}}([\lambda]) by Eq. (73), is a polynomial in $S$ and $Q$, this then holds also for $S>Q$. As a result, when $v_{\text{R}}$ and $S$ are taken as the independent variables, a term $-2S(S+1)$ cancels out in the difference C_{\text{\mathfrak{gl}(d)}}([\lambda^{\prime}])-C_{\text{\mathfrak{o}(d)}}([\lambda]), where $[\lambda^{\prime}]$ is the enclosing $\mathfrak{gl}(d)$ irrep. This explains that the eigenvalue of the pairing interaction considered in Ref. Racah, 1943 depends only on $n$ and $v_{\text{R}}$ and not on $S$ and $\sigma$, cf. Eq. (50) there. When $v$ rather than $v_{\text{R}}$ is taken as the independent valiable, this eigenvalue depends on $S$ and $\sigma$ in a way analogous to the dependence of such an eigenvalue on isospin and reduced isospin in the system of neutrons and protons sharing a configuration space.Edmonds and Flowers (1952)

The algebra $\mathfrak{o}(3)_{S}$ is identical to the algebra $\mathfrak{sl}(2)$ of operators $Y$ in Eq. (70) with a traceless $y$. For $k>2$ one may similarly decompose the $\mathfrak{o}(2k)$ representation into $\mathfrak{sl}(k)$ irreps, which may be combined with the $\mathfrak{o}(d)$ representation to classify basic states with a fixed $n$. The $\mathfrak{o}(2k)$ algebra then connects such representation spaces with different $n$. For $k>2$ the $\mathfrak{o}(2k)$ algebra is simple, so the $\mathfrak{sl}(k)$ algebra has no commuting subalgebra like $\mathfrak{o}(3)_{Q}$. Therefore, while the connected representations have equal $\mathfrak{o}(d)$ factors, their $\mathfrak{sl}(k)$ factors may differ.

As an example, the $\mathfrak{sl}(4)$ subalgebra of the $\mathfrak{o}(8)$ algebra of Flowers and Szpikowski was identified by Wigner in the early days of nuclear theory. Wigner (1937) Traditionally, it is referred to by nuclear physicists as $SU(4)$, and its irreps are decomposed further into irreps of a direct sum $\mathfrak{sl}(2)_{S}\oplus\mathfrak{sl}(2)_{T}$ of Lie algebras of spin and isospin. Kota and Castilho Alcarás determined the branching of $\mathfrak{o}(8)$ representations into $\mathfrak{sl}(4)$, or $\mathfrak{gl}(4)$, irreps by analyzing the dual branching of $\mathfrak{gl}(d)$ irreps into $\mathfrak{o}(d)$ representations. Kota and Castilho Alcarás (2006)

The discussion above of systems of fermions with spin $s=1/2$ may be generalized to fermions with $s>1/2$. One then has $k=2s+1>2$. Because fermion spins are half-integral, $k$ is even. In principle a symmetric metric $g$ may be defined on any 1-particle configuration space, including configuration spaces of single fermions. But in the fermion case, unlike the Wigner metric (4), $g$ will not be rotationally invariant. The Wigner metric (58) on a space of 1-fermion states is antisymmetric and so gives rise to a symplectic Lie algebra. This lack of rotational invariance seems to bar the relevance of the $\mathfrak{o}$-$\mathfrak{o}$ duality to systems with an odd number of fermion kinds (more than one) such as the three colors or the three low-mass flavors of quarks.

VIII Summary

The Fock space $\Phi$ spanned by all possible states of any number of fermions of $k$ different kinds inhabiting a common configuration space of dimension $d$ is considered. The theorem of orthogonal-orthogonal duality states that $\Phi$ is composed of outer products $\text{X}_{[\lambda]}\otimes\Psi_{[\mu]}$, where $\text{X}_{[\lambda]}$ carries the representation of $\mathfrak{o}(d)$ associated with the partition $[\lambda]$, and $\Psi_{[\mu]}$ similarly, involving the Lie algebra $\mathfrak{o}(2k)$. Here $\mathfrak{o}(d)$ denotes the Lie algebra of infinitesimal orthogonal transformations of a vector space of dimension $d$. Each pair of partitions $[\lambda]$ and $[\mu]$ which obey a certain criterion appears exactly once. This criterion can be expressed in a simple geometric form: The $[\lambda]$ Young diagram and a reflected copy of the $[\mu]$ Young diagram fill a rectangle of width $k$ and height $d/2$ without overlap. The algebra $\mathfrak{o}(d)$ conserves the number of fermions in the system and the algebra $\mathfrak{o}(2k)$ does not.

An analogous symplectic-symplectic duality was proven by Helmers almost 60 years ago, but only recently was the orthogonal-orthogonal duality proven by a very different method. Rowe, Repka, and Carvalho (2011) I have presented a proof by the same method as applied by Helmers, using the known expressions for the characters of orthogonal Lie algebras.

Besides the orthogonal-orthogonal and symplectic-symplectic dualities, the authors of Ref. Rowe, Repka, and Carvalho, 2011 list and prove a unitary-unitary duality. This expresses the following fact, known since long ago: The space of antisymmetric functions of $n$ pairs of variables $(p,\tau)$, where $p$ takes $d$ values and $\tau$ takes $k$ values, is composed of antisymmetrized products of a function $\psi$ of the variables $p$ and a function $\chi$ of the variables $\tau$. In this direct sum, each antisymmetrized product of an irrep of the general linear Lie algebra $\mathfrak{gl}(d)$ (or its unitary subalgebra $\mathfrak{u}(d)$) acting on $\psi$, and an irrep of the general linear Lie algebra $\mathfrak{gl}(k)$ (or its unitary subalgebra $\mathfrak{u}(k)$) acting on $\chi$, appear exactly once according to the criterion that the two irreps must be associated with Young diagrams that are reflections of one another.

In the present article, the geometric relation between the Young diagrams involved in each of these three duality theorems was shown to allow very simple derivations of linear relations between the Casimir invariants of the connected representations. Only particular cases of these linear relations seem to have been presented previously in the literature.

A final section addressed the question of possible applications the orthogonal-orthogonal duality to the analysis of actual physical systems other than such already considered in the literature.