Component Decompositions and Adynkra Libraries for Supermultiplets in Lower Dimensional Superspaces

We present Adynkra Libraries that can be used to explore the embedding of multiplets of component field (whether on-shell or partial on-shell) within Salam-Strathdee superfields for theories in dimension nine through four.


Introduction
In a continuing series of works [1,2,3], we have been developing a new approach to supersymmetry and supermultiplets that possesses the clarity of traditional component field approaches without their incompleteness. Simultaneously this new approach also retains the completeness of superfield approaches without their opaqueness. Our approach, likely applicable in all dimensions 4 , has led to the construction of "adynkrafields" [3]. Adynkrafields are based on adinkra graphs [4] and Dynkin Labels [5,6].
An adynkrafield [3] may be obtained by starting with a traditional Salam-Strathdee superfield [7] and replacing the familiar Grassmann coordinate expansions required to derive component results by two distinct types of special configurations of Young Tableaux (YT's) to derive component results. Both sets of quantities (θ-monomials versus specialized YT combinations) for expansions, respectively, are distinct sets of "basis vectors" for the description of component fields within offshell supermultiplets.
However, it is important to realize that taking a starting point that solely begins with the considerations of Dynkin Labels, YT's and their properties is logically viable and independent of the concept of the Salam-Strathdee superfield. This means that as far as the representation theory of supersymmetry and supermultiplets go, the latter can be entirely replaced by ordinary concepts that exist in Lie Algebra theory without reference to the superfield concept.
The algorithmically superiority of YT-expansions over θ-expansions can be understood by an analogy. It is well known in physics, when a problem has spherical symmetry (take for example the quantum mechanical hydrogen atom) the superior calculational direction is to employ spherical (r, θ, φ) coordinates 5 , not rectilinear (x, y, z) ones. Taking the path of using the triplet (r, θ, φ) coordinates leads to the discovery of solutions in terms of "hydrogenic wavefunctions." If one is engaged in finding numerical solutions for atomic physics problems, efficient algorithms begin with the use of "hydrogenic wavefunctions." Thus, we assert if one is engaged in finding component results for supersymmetric problems, efficient algorithms begin with the use of adynkrafields, not superfields. The most powerful demonstration of this was given in the work seen in reference [3].
Continuing to use this analogy, when one examines the eigenfunctions of the Laplace operator in three spatial dimensions, the spherical harmonics naturally emerge as the angular portion of it solutions. On the other hand, for the Laplace operator in two spatial dimensions the angular portion of the solution is dominated by complex exponentials. Of course, complex exponentials are a part of the spherical harmonics. This phenomenon of the emergence of a lower dimensional set of orthogonal function under a dimensional reduction is the main point of this current work.
In this approach, all component fields are reduced to two pieces of data, the corresponding Dynkin Label, and the corresponding engineering height. The first piece of data encodes the Lorentz representation of the field and the second controls how a component field can appear in an action. In the construction of adynkras, these two pieces of data are quite simple to track. Thus when one is examining the question of whether there exists a superfield into which any boson-fermion 4 The realm in which our explicit constructions and discussions have been presented are for 10D and 11D supersymmetrical systems. 5 In this triple of coordinates, of course, θ denotes the polar angle, not the Grassmann coordinate.
pair can be embedded, on a conceptual basis, the problem becomes identical to one where partial information concerning genetic (DNA/RNA) content ("a snippet") is known and the question is one of finding the genetic sequence to which the snippet belongs. Another way to describe this is as a "pattern-matching" problem. We are thus in the position to create efficient algorithms capable of answering such questions about the embedding of component fields into superfields.
We wish to show how the adynkras of a theory in a lower spatial dimension emerge from those of a higher dimension. Our starting point will be the adynkra for eleven dimensions. Since we have previously given the answer which emerges from the reduction to nine spatial dimensions, the new results in this work will concentrate on the cases of 4 ≤ D ≤ 9. For each of these cases, we will derive the unconstrained scalar superfield adynkras that emerge from the reduction from a higher dimension.
In Chapter two, we briefly review the construction of superspaces in various spacetime dimensions. Two approaches used to construct adynkrafields are reviewed as well. From Chapter three to Chapter eight, we present the following from 9D to 4D respectively, Note that (e.) only applies to 8D and to 5D.
In each of the dimensions M of the associated Minkowski space under consideration, the adynkra V M associated with the scalar superfield is explicitly presented. This corresponds in every case to a library that explicitly lists the Lorentz representations of all the component fields contained in the scalar superfield appropriate to that dimension. Given the Dynkin Label description of any bosonic irrep (denoted by R) or any fermionic irrep (denoted by R), the quantities V R = R ⊗ V M and V R = R ⊗ V M respective describe the Lorentz component field contents of any superfield over the Minkowski space of dimension M.
The content of the typical V M -library is a listing of Dynkin Labels and the associated heights at which the corresponding irreps appear. For some heights, it is seen that a given Dynkin Label appears with a frequency greater than one. This same phenomenon has been noted previously in the work of [8] where Dynkin Labels have been extensively used to describe the component field content of 11D superfields 6 .
Chapter nine presents our conclusions.
There are two appendices included in this work. Appendix A contains a dictionary between Dynkin Labels and the corresponding representation dimensionalities of Lorentz groups in various spacetime dimensions. Appendix B is devoted to present all necessary projection matrices for the branching rules used to develop minimal scalar superfields and (1, 0) superfields in various spacetime dimensions.
The final portion of this paper includes our references. 6 In particular this is seen in the equation listed as (2.7) in this reference.

Methodology
A superspace consists of a set of spacetime coordinates and spinor coordinates. For each spacetime dimension D, there is a specific type of spinors and a minimal number of real components of the spinor coordinates that we call d. This information is listed in Table 1. We use the Minkowski signature (−, +, +, . . . , +) in every dimension, and the corresponding Lorentz group is SO(1, D − 1).

Spacetime Dimension Lorentz Group
Type of Spinors d 11 SO(1,10) Majorana 32 10 SO (1,9) Majorana-Weyl 16 9 SO (  In our previous research work [1,2], we decomposed each level of the 10D, N = 1, N = 2A, N = 2B, and 11D, N = 1 scalar superfields into Lorentz representations. A scalar superfield with d spinor coordinates can be Taylor expanded into (d + 1) levels -from level-0 (zeroth power of spinor coordinates) to level-d (d-th power of spinor coordinates). In an arbitrary level-n, the decomposition of θ-monomials translates to the antisymmetric product of n spinor representation {d} in so(1, D − 1), which we denote by [{d} ⊗ n ] A or {d} ∧ n . The total degrees of freedom at level-n is the binomial coefficient ( d n ). Note that this is true only for unconstrained scalar superfields. Two methods were used in obtaining the results of the antisymmetric products of each level in terms of direct sums of Lorentz irreducible representations. One utilizes the branching rules su(d) ⊃ so(1, D − 1) of the totally antisymmetric irreps in su(d) constructed by the fundamental representation {d}. These calculations can be accomplished by either Susyno [10] or LieART [11]. Both of them are availble online 7 . The relevant projection matrices used are listed in Appendix B. The other one involves the concept of plethysms, and the calculations are done by Susyno [10]. The details of these methods are explained explicitly in Chapter 4 of [2].
In [3], there is a complete methodological discussion on how to interpret both the bosonic and the spinorial irreps of so(1, D − 1), so that we can translate the scalar superfield components from Dynkin labels to Young Tableaux to field variables, and obtain the corresponding irreducible conditions. One can draw adynkra diagrams to encode all the information of a superfield. The links in the diagrams indicate sufficient but not necessary supersymmetry transformations between component fields, and the algorithm is described in Section 8.3 of [3]. These techniques are used in this paper.

Component Decompostion Results
The 9D minimal superfield component decomposition results by Dynkin Labels are shown below.
The component content of 9D minimal scalar superfield (up to  is summarized in the Adynkra diagram in Figure 1 and Adinkra diagram in Figure 2. The definitions of Adinkra diagrams in spacetime dimension larger than one and Adynkra diagrams were presented in [1,3]. In order to establish a graphical language to describe bosonic irreducible representations in so(9), we define irreducible bosonic Young Tableaux as what we did in 10D [3].
Consider the projection matrix for su(9) ⊃ so(9) [9], P su (9) (3.1) The highest weight of a specified irrep of su(9) is a row vector [p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 7 , p 8 ], where p 1 to p 8 are non-negative integers. Since the su(9) YT with n vertical boxes is the conjugate of the one with 9 − n vertical boxes, we need only consider the p 5 = p 6 = p 7 = p 8 = 0 case.

(3.2)
Thus, given an irreducible bosonic Young Tableau with p 1 columns of one box, p 2 columns of two vertical boxes, p 3 columns of three vertical boxes, and p 4 columns of four vertical boxes, the Dynkin Label of its corresponding bosonic irrep is [p 1 , p 2 , p 3 , 2p 4 ]. Then look at the congruence classes of a representation with Dynkin Label [a, b, c, d] in so (9), C c (R) classifies the bosonic irreps and spinorial irreps: C c (R) = 0 is bosonic and C c (R) = 1 is spinorial. Consequently, a bosonic irrep satisfies d = 0 (mod 2).
Therefore, we can reverse this process and show the one-to-one correspondence between bosonic Dynkin Label irreps and irreducible bosonic Young Tableaux. Namely, given a bosonic irrep with Dynkin Label [a, b, c, d], its corresponding irreducible bosonic Young Tableau is composed of a columns of one box, b columns of two vertical boxes, c columns of three vertical boxes, and d/2 columns of four vertical boxes.
The simplest examples, also the fundamental building blocks of a BYT, are given below. The YT's with "IR" subscript subject to certain conditions refer to irreducible representations of so (9). YT's without this subscript are reducible with respect to so(9) while irreducible with respect to su (9). The irreducibility conditions are effectuated by the branching rules for su(9) ⊃ so (9), which can also be derived by "tying rules" invented in [3].
For spinorial irreps, the basic SYT is given by (3.5) We could translate the Dynkin Label of any spinorial irrep to a mixed YT (which contains a BYT part and a basic SYT above) with irreducible conditions by applying the same idea discussed in Chapter five of [3].
Putting together the columns in (3.4) and (3.5) corresponds to adding their Dynkin Labels.
In summary, the irreducible Young Tableau descriptions of the 9D minimal scalar superfield decomposition is presented below.
(3.6) where Level-9 to Level-16 have exactly the same expressions as Level-7 to Level-0.

Index Structures and Irreducible Conditions of Component Fields in 9D Minimal Scalar Superfield
In this section, we will translate the irreducible bosonic and mixed Young Tableaux into field variables. In order to express the index structure clearly and efficiently, we introduce the following notational conventions for irreducible bosonic YT. We use "|" to separate indices in YT with different heights and "," to separate indices in YT with the same heights.
The vector index a runs from 0 to 8. In fact, the {}-indices, irreducible bosonic Young Tableaux, and Dynkin Labels are equivalent and have the one-to-one correspondence.
The general expression is as below in Equations (3.7) and (3.8), where above we have "disassembled" the YT to show how each column is affiliated with each type of subscript structure. Below we have assembled all the columns into a proper YT.  As one moves from the YT's shown in Equation (3.7) to Equation (3.8), it is clear the number of vertical boxes is tabulating the number of 1-forms, 2-forms, 3-forms, and 4-forms in the YT's. These are the entries between the vertical | bars. These precisely correspond to the integers p, q, r, and s in the Dynkin Labels. An example of the correspondence between the subscript conventions, the affiliated YT, and Dynkin Label is shown in (3.9). (3.9) The index structures as well as irreducible conditions of all bosonic and fermionic fields are identified below along with the level at which the fields occur in the adinkra of the superfield. The spinor index α runs from 1 to 16.
Level-9 to Level-16 have exactly the same expressions as Level-7 to Level-0.

Component Decompostion Results
The 8D minimal superfield component decomposition results in terms of Dynkin Labels are shown below.
The component decompostion results by dimensions are shown below.
The Adynkra and Adinkra diagrams for 8D minimal scalar superfield (up to  are Figure 3   (4.1) The highest weight of a specified irrep of su (8) is a row vector [p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 7 ], where p 1 to p 7 are non-negative integers. Since the su(8) YT with n vertical boxes is the conjugate of the one with 8 − n vertical boxes, we need only consider the p 5 = p 6 = p 7 = 0 case.

(4.2)
Note that in 8D, duality condition needs to be considered for the four-form. Similar as the situation in 10D discussed in [3], the Dynkin Label [p 1 , p 2 , p 3 + 2p 4 , p 3 ] carries the same dimensionality and corresponds to the same YT shape as [p 1 , p 2 , p 3 , p 3 + 2p 4 ], although there's no complex conjugation in so (8) and all irreps are real. Then look at the congruence classes of a representation with Dynkin Label [a, b, c, d] in so (8), Based on the above equations, there are totally four congruence classes in so (8), .
Thus, given an irreducible bosonic Young Tableau with p 1 columns of one box, p 2 columns of two vertical boxes, p 3 columns of three vertical boxes, and p 4 columns of four vertical boxes, the Dynkin Label of its corresponding bosonic irrep is [p 1 , p 2 , p 3 , p 3 + 2p 4 ] (or [p 1 , p 2 , p 3 + 2p 4 , p 3 ]). Reversely, given a bosonic irrep with Dynkin Label [a, b, c, d], its corresponding irreducible bosonic Young Tableau is composed of a columns of one box, b columns of two vertical boxes, min{c, d} columns of three vertical boxes, and |d − c|/2 columns of four vertical boxes.
The simplest examples, also the fundamental building blocks of an irreducible BYT, are given below.
For spinorial irreps, the basic SYTs are given by We could translate the Dynkin Label of any spinorial irrep to a mixed YT (which contains a BYT part and one of the basic SYT above) with irreducible conditions by applying the same idea discussed in Chapter five in [3].
Putting together the columns in (4.5) and (4.6) corresponds to adding their Dynkin Labels. We follow the well-known (VCS) convention for labelling so(8) irreps to name our irreducible YTs.
In summary, the irreducible Young Tableau descriptions of the 8D minimal scalar superfield decomposition is presented below.

Index Structures and Irreducible Conditions of Component Fields in 8D Minimal Scalar Superfield
In this section, we will translate the irreducible bosonic and mixed Young Tableaux into field variables. We follow the same {}-indices notation as well as "|" to separate indices in YT with different heights and "," to separate indices in YT with the same heights.
The vector index a runs from 0 to 7. The {}-indices, irreducible bosonic Young Tableaux, and Dynkin Labels are equivalent and have the one-to-one correspondence.
The general expression is as below in Equations (4.7) and (4.8), where above we have "disassembled" the YT to show how each column is affiliated with each type of subscript structure. Below we have assembled all the columns into a proper YT.
As one moves from the YT's shown in Equation (4.7) to Equation (4.8), it is clear the number of vertical boxes is tabulating the number of 1-forms, 2-forms, 3-forms, and 4-forms in the YT's. These are the entries between the vertical | bars. These precisely correspond to the integers p, q, r, and s that appeared in Dynkin Labels. An example of the correspondence between the subscript conventions, the affiliated YT, and Dynkin Label is shown in (4.9). (4.9) In 8D, we have two types of spinor indices corresponding to 8s and 8c respectively. We define the field Ψ α or Ψ . α correpsonds to 8s and the field Ψ α or Ψ .
The index structures as well as irreducible conditions of all bosonic and fermionic fields are identified below along with the level at which the fields occur in the adinkra of the scalar superfield.
βαβ (x) + · · · + · · · (4.10) which implies that for example, Level-2 in the scalar superfield decomposition is nothing but Last but not least, we can draw the adynkra and adinkra diagrams corresponding to the (1, 0) multiplet, which are Figures 5 and 6.

Component Decompostion Results
The 7D minimal superfield component decompostion results by Dynkin Labels are given below.

7D Minimal Adinkra Diagram
The Adynkra and Adinkra diagrams for 7D minimal scalar superfield (up to  are shown in Figures 7 and 8.

Young Tableaux Descriptions of Component Fields in 7D Minimal Scalar Superfield
Consider the projection matrix for su(7) ⊃ so(7) [9], P su (7) The highest weight of a specified irrep of su (7) is a row vector [p 1 , p 2 , p 3 , p 4 , p 5 , p 6 ], where p 1 to p 6 are non-negative integers. Since the su (7) YT with n vertical boxes is the conjugate of the one with 7 − n vertical boxes, we only need to consider the p 4 = p 5 = p 6 = 0 case.  Starting from the weight vector [p 1 , p 2 , p 3 , 0, 0, 0] in su (7), we define its projected weight vector [p 1 , p 2 , 2p 3 ] in so (7) as the Dynkin Label of the corresponding irreducible bosonic Young Tableau.

(5.2)
Then look at the congruence classes of a representation with Dynkin Label [a, b, c] in so (7), We could translate the Dynkin Label of any spinorial irrep to a mixed YT (which contains a BYT part and a basic SYT above) with irreducible conditions by applying the same idea discussed in Chapter five in [3].
Putting together the columns in (5.4) and (5.5) corresponds to adding their Dynkin Labels.
In summary, the irreducible Young Tableau descriptions of the 7D minimal scalar superfield decomposition is presented below.
(5.6) where Level-9 to Level-16 have exactly the same expressions as Level-7 to Level-0.

Index Structures and Irreducible Conditions of Component Fields in 7D Minimal Scalar Superfield
In this section, we will translate the irreducible bosonic and mixed Young Tableaux into field variables. We follow the same {}-indices notation as well as "|" to separate indices in YT with different heights and "," to separate indices in YT with the same heights.
The vector index a runs from 0 to 6. The {}-indices, irreducible bosonic Young Tableaux, and Dynkin Labels are equivalent and have the one-to-one correspondence.
The general expression is as below in Equations (5.7) and (5.8), where above we have "disassembled" the YT to show how each column is affiliated with each type of subscript structure. Below we have assembled all the columns into a proper YT.  The index structures as well as irreducible conditions of all bosonic and fermionic fields are identified below along with the level at which the fields occur in the adinkra of the scalar superfield. The spinor index α runs from 1 to 8.

(1, 0) Multiplet Decompositions
In seven spacetime dimensions, although we don't have chiral spinors, instead we have two copies of [0, 0, 1] due to the SU(2)-Majorana condition. Starting from one copy of [0, 0, 1] spinor, we can construct a supermultiplet which is the subset of the one constructed from the minimal scalar superfield. We call it as (1, 0) multiplet decomposition. Apply the Plethysm function and results are as below. Using branching rules for su(8) ⊃ so (7) gives the same results. The projection matrix is presented in Equation (B.8). Starting from the above decompositions, we can reproduce the minimal scalar superfield decompositions using the similar idea as when we construct 10D Type IIB scalar superfield from 10D Type I superfield. Basically we can label spinors as θ α and θ α both corresponding to the same irrep [0, 0, 1]. Then expand the scalar superfield only with respect to θ α first. Namely, αβαβ (x) + · · · + · · · (5.10) which implies that for example, Level-2 in the scalar superfield decomposition is nothing but Last but not least, we can draw the adynkra and adinkra diagrams corresponding to the (1, 0) multiplet, which are Figures 9 and 10.     The highest weight of a specified irrep of su (6) is a row vector [p 1 , p 2 , p 3 , p 4 , p 5 ], where p 1 to p 5 are non-negative integers. Since the su(6) YT with n vertical boxes is the conjugate of the one with 6 − n vertical boxes, we only need to consider the p 4 = p 5 = 0 case.
Given an irreducible bosonic Young Tableau with p 1 columns of one box, p 2 columns of two vertical boxes, and p 3 columns of three vertical boxes, the Dynkin Label of its corresponding bosonic irrep is [p 1 , p 2 , p 2 + 2p 3 ] (or [p 1 , p 2 + 2p 3 , p 2 ]). Reversely, given a bosonic irrep with Dynkin Label [a, b, c], its corresponding irreducible bosonic Young Tableau is composed of a columns of one box, min{b, c} columns of two vertical boxes, and |c − b|/2 columns of three vertical boxes.
The simplest examples, also the fundamental building blocks of an irreducible BYT, are given below. We could translate the Dynkin Label of any spinorial irrep to a mixed YT (which contains a BYT part and one of the basic SYTs above) with irreducible conditions by applying the same idea discussed in Chapter five in [3].
Putting together the columns in (6.5) and (6.6) corresponds to adding their Dynkin Labels.
In summary, the irreducible Young Tableau descriptions of the 6D minimal scalar superfield decomposition is presented below. (6.7)

Index Structures and Irreducible Conditions of Component Fields in 6D Minimal Scalar Superfield
In this section, we will translate the irreducible bosonic and mixed Young Tableaux into field variables. We follow the same {}-indices notation as well as "|" to separate indices in YT with different heights and "," to separate indices in YT with the same heights.
The vector index a runs from 0 to 5. The {}-indices, irreducible bosonic Young Tableaux, and Dynkin Labels are equivalent and have the one-to-one correspondence.
The general expression is as below in Equations (6.8) and (6.9), As one moves from the YT's shown in Equation (6.8) to Equation (6.9), it is clear the number of vertical boxes is tabulating the number of 1-forms, 2-forms, and 3-forms in the YT's. These are the entries between the vertical | bars. These precisely correspond to the integers p, q, and r appeared in Dynkin Labels. An example of the correspondence between the subscript conventions, the affiliated YT, and Dynkin Label is shown in (6.10). In 6D, we have two types of spinor indices corresponding to 4 and 4 respectively. We define the field Ψ α or Ψ . α correpsonds to 4 and the field Ψ α or Ψ .
The index structures as well as irreducible conditions of all bosonic and fermionic fields are identified below along with the level at which the fields occur in the adinkra of the superfield.
Level-5 to Level-8 have exactly the same expressions as Level-3 to Level-0.    The highest weight of a specified irrep of su(5) is a row vector [p 1 , p 2 , p 3 , p 4 ], where p 1 to p 4 are non-negative integers. Since the su(5) YT with n vertical boxes is the conjugate of the one with 5 − n vertical boxes, we need only consider the p 3 = p 4 = 0 case.
Starting from the weight vector [p 1 , p 2 , 0, 0] in su (5), we define its projected weight vector [p 1 , 2p 2 ] in so(5) as the Dynkin Label of the corresponding irreducible bosonic Young Tableau.
For spinorial irreps, the basic SYT is given by We could translate the Dynkin Label of any spinorial irrep to a mixed YT (which contains a BYT part and a basic SYT above) with irreducible conditions by applying the same idea discussed in Chapter five in [3].
Putting together the columns in (7.4) and (7.5) corresponds to adding their Dynkin Labels.
In summary, the irreducible Young Tableau descriptions of the 5D minimal scalar superfield decomposition is presented below. In this section, we will translate the irreducible bosonic and mixed Young Tableaux into field representation. We follow the same {}-indices notation as well as "|" to separate indices in YT with different heights and "," to separate indices in YT with the same heights.
The vector index a runs from 0 to 4. The {}-indices, irreducible bosonic Young Tableaux, and Dynkin Labels are equivalent and have the one-to-one correspondence.
The general expression is as below in Equations (7.7) and (7.8), where above we have "disassembled" the YT to show how each column is affiliated with each type of subscript structure. Below we have assembled all the column into a proper YT.
As one moves from the YT's shown in Equation (7.7) to Equation (7.8), it is clear the number of vertical boxes is tabulating the number of 1-forms and 2-forms in the YT's. These are the entries between the vertical | bars. These precisely correspond to the integers p and q appeared in Dynkin Labels. An example of the correspondence between the subscript conventions, the affiliated YT, and Dynkin Label is shown in (7.9).
The index structures as well as irreducible conditions of all bosonic and fermionic fields are identified below along with the level at which the fields occur in the adinkra of the superfield. The spinor index α runs from 1 to 4. In five spacetime dimensions, the story is pretty similar as in seven dimensions since in both cases SU(2)-Majorana condition plays an important role. Consequently we have two copies of [0, 1]. Start from one copy of [0, 1] spinor, we can also construct a supermultiplet called (1, 0) multiplet which is the subset of the one constructed from the scalar superfield. Apply the Plethysm function and results are as below. Using branching rules for su(4) ⊃ so (5) gives the same results. The projection matrix is presented in Equation (B.10). Starting from the above decomposition and we can reproduce the scalar superfield decomposition using the similar idea as in 7D. Basically we can label spinors as θ α and θ α both corresponding to the same irrep [0, 1]. Then expand the scalar superfield only with respect to θ α first. For example, Level-2 in the scalar superfield decomposition is nothing but ( Last but not least, we can draw the adynkra and adinkra diagrams corresponding to the (1, 0) multiplet, which are Figures 17 and 18.

Component Decompostion Results
The 4D minimal scalar superfield component decompostion results by Dynkin Labels is given as follows. Here we also list the component decompostion results by dimensions.
In Table 1  The highest weight of a specified irrep of su(4) is a row vector [p 1 , p 2 , p 3 ], where p 1 to p 3 are nonnegative integers. Since the su(4) YT with n vertical boxes is the conjugate of the one with 4 − n vertical boxes, we only need to consider the p 3 = 0 case.
Given an irreducible bosonic Young Tableau with p 1 columns of one box and p 2 columns of two vertical boxes, the Dynkin Label of its corresponding bosonic irrep is [p 1 , p 1 + 2p 2 ] (or [p 1 + 2p 2 , p 1 ]). Reversely, given a bosonic irrep with Dynkin Label [a, b], its corresponding irreducible bosonic Young Tableau is composed of min{a, b} columns of one box and |b − a|/2 columns of two vertical boxes.
The simplest examples, also the fundamental building blocks of an irreducible BYT, are given below. In summary, if we adopt the 4-component Majorana notation, the irreducible Young Tableau descriptions of the 4D minimal scalar superfield decomposition can be presented as follows. (8.5)

Index Structures and Irreducible Conditions of Component Fields in 4D Minimal Scalar Superfield
In this section, we will translate the irreducible bosonic Young Tableaux into field representation. We follow the same {}-indices notation as well as "|" to separate indices in YT with different heights and "," to separate indices in YT with the same heights.
The vector index a runs from 0 to 3. The {}-indices, irreducible bosonic Young Tableaux, and Dynkin Labels are equivalent and have the one-to-one correspondence.
The general expression is as below in Equations (8.6) and (8.7), As one moves from the YT's shown in Equation (8.6) to Equation (8.7), it is clear the number of vertical boxes is tabulating the number of 1-forms and 2-forms in the YT's. These are the entries between the vertical | bars. These precisely correspond to the Dynkin Labels p and q. An example of the correspondence between the subscript conventions, the affiliated YT, and Dynkin Label is shown in (8.8).
The index structures as well as irreducible conditions of all bosonic and fermionic fields are identified below along with the level at which the fields occur in the adinkra of the superfield. The Majorana spinor index α runs from 1 to 4.

Conclusion
In this work, we have established the basic libraries of adynkras that can be used to explore problems of embedding component fields into superfields in the context of spacetimes with Lorentzian signature and D − 1 spatial dimensions where 4 ≤ D ≤ 9. In addition to minimal scalar superfields, we also explored (1, 0) multiplets in several dimensions.
How to write the complete set of supersymmetry transformations for these component fields is an important question. In every adynkra diagram we draw, the supersymmetry transformations are determined by a sufficient but not neccesary condition as stated in Section 8.3 of [3]. A fully satisfactory answer would require the construction of supercovariant derivative operators acting on adynkrafields. We will discuss the 4D case in a paper in preparation and its generalizations deserve further study.
The importance of the adynkra libraries is that for the first time in the history of the subject of supersymmetry, these will support the creation of algorithms that solve the embedding problem. One can now start with any spectrum of on-shell component fields in these various dimensions and algorithmically derive the minimal dimension superfield representation (as well as alternatives) that contains this specified component field spectrum. The technique for this has been christened as an"adynkra digital analysis" (ADA) scan as described in the the work of [13].
"I am thinking about something much more important than bombs. I am thinking about computers." -John von Neumann   Table 9: so(4) irreducible representations [9]

B Projection Matrices for Branching Rules
Here we list the projection matrices for the branching rules we used to obtain both the Lorentz component contents of minimal scalar superfields and (1, 0) superfields from 9D to 5D.

B.1 Minimal Scalar Superfield Decompositions
The relevant branching rules for decomposing minimal scalar superfields are su(d) ⊃ so(D), where d is the number of real components of Grassman coordinates and D is the spacetime dimension.
Recall the definition of a projection matrix P g⊃h , v T h = P g⊃h w T g , (B.13) where w g and v h are weight vectors in g and h respectively. In the discussion of so(8) BYTs, we want the fundamental representation {8} in su (8) projected to the vector in so (8) su (8)⊃so (8) . One quick way to distinguish between two projection matrices is to look at their first column -if it is [1, 0, 0, 0] T then it is (vector), and if it is [0, 0, 0, 1] T then it is (spinor).
Similarly, in the discussion of so(4) BYTs, we project the fundamental representation {4} in su(4) to the vector in so(4), i.e. su(4)⊃so (4) . We conclude by saying that the coincidence of both vector and spinor having d.o.f. = 8 in 8D, or the triality of so (8), creates the opportunity of writing two P su(8)⊃so(8) matrices while we think of them as descending from the same fundamental irrep {8} in su (8), since there is no vectorial / spinorial distinction of irreps in su (8). The same is true for su(4) ⊃ so(4), as both vector and spinor in 4D have a total number of 4 components. Therefore, two versions of P su(8)⊃so (8) serve two different purposes, and so do the two versions of P su(4)⊃so (4) , and there is no contradiction.

B.4 A Note on Isomorphisms
In [9] and [10], the discussions or the programming tools for calculations of some particular algebras such as so(6), so(5) and so(4) are absent. This is because they are isomorphic to some other simple lie algebras,