DETECTING ALGEBRA OBJECTS FROM NIM-REPS IN POINTED, NEAR-GROUP AND QUANTUM GROUP-LIKE FUSION CATEGORIES

. In this article we study the possible Morita equivalence classes of algebras in three families of fusion categories (pointed, near-group and p A 1 , l q 12 ) by studying the Non-negative Integer Matrix representations (NIM-reps) of their underlying fusion ring, and compare these results with existing classiﬁcation results of algebra objects. Also, in an appendix we include a test of the exponents conjecture for modular tensor categories of rank up to 4


Introduction
The study of fusion rings and their associated non-negative integer matrix representations (or NIM-reps, for short) has stimulated a rich production in the literature, see e.g.[BPPZ00, DFZ90, EP21, Gan02, Gan05].A good reason behind this is the connection of NIM-reps to boundary rational conformal field theory (or D-branes in string theory): in particular, finding NIM-reps is equivalent to solving Cardy's equation in these cases.Some recent papers have continued this work [CRSS23, YZL1806], showing that this interest keeps up after time.
Given a rational, c 2 -cofinite vertex operator algebra describing the chiral symmetries of a rational conformal field theory, its category of representations is a modular tensor category [Hua08,HLZ].In this paper we go for a slightly more general setting than modular and focus on the study and .Furthermore, there is an interesting conjecture relating the exponents of NIM-reps and those of the modular invariants of the category they live on.In an appendix, we test explicitly this conjecture for unitary modular tensor categories up to rank 4 (which by [RSW09] include examples of the three families of categories considered in this paper for which we compute their respective NIM-reps).
The structure of the paper is as follows.In Section 2 we introduce all the necessary background on fusion and module categories as well as algebra objects, and also on Z `-modules and NIM-reps.In Section 3 we compute the NIM-reps for each category, and also the algebra objects derived from those.Appendix A includes the calculations testing the exponents conjecture for modular tensor categories of rank up to 4.
Acknowledgements.SH is supported by the Engineering and Physical Sciences Research Council.ARC is supported by Cardiff University.Devi Young's contribution was supported by the Cardiff University On-Campus Internship Scheme 21/22.The authors would like to especially thank an anonymous referee for their helpful comments and suggestions.

Preliminaries
Throughout this paper, we fix ❦ to be an algebraically closed field.In this section, we collect some basic definitions necessary for our work.
2.1.Group Actions.Here we recall several standard definitions and well-known results on group actions, e.g.see [Arm10,Cam99].
Definition 2.1.Let G be a group and S a set.A G-action on S is a binary operation ˚: G ˆS Ñ S such that, for all s P S, e ˚s " s, and pg ¨hq ˚s " g ˚ph ˚sq, where e P G is the group identity element.A set with such an action is called a G-set.
Definition 2.2.Let S be a G-set, and s P S.
-The orbit of G through s is the subset of S defined by Orbpsq " tg ˚s|g P Gu.
-The G-action on S is called transitive if Orbpsq " S.
-The stabiliser of s is the subgroup of G defined by Stabpsq " tg P G|g ˚s " su.
For two elements s, t P S, the orbits Orbpsq, Orbptq are either equal or disjoint.Hence the set S can be partitioned into a collection of transitive G-sets.The following results can be found at [Cam99,Theorem 1.3].
Proposition 2.3.Let S be a G-set, and take s P S.There is an isomorphism of G-sets between the orbit Orbpsq and the set of left cosets G{Stabpsq, where G acts on the set of left cosets by left-translation.
Proposition 2.4.Two left coset G-sets G{H,G{K are isomorphic as G-sets if and only if H, K are conjugate as subgroups of G.
As a result of these propositions, we can study all G-sets by studying the G-sets of left cosets for all conjugacy classes of subgroups in G.

Fusion Categories.
In this subsection we follow [EGNO15].
A monoidal category C consists of a tuple pC, b, ✶, α, l, rq where C is a category, b : C ˆC Ñ C is a bifunctor, ✶ P Ob pCq, α X,Y,Z : pX b Y q b Z Ñ X b pY b Zq is a natural isomorphism for each X, Y, Z P Ob pCq, and l X : ✶ b X Ñ X and r X : X b ✶ Ñ X are natural isomorphisms for all X P Ob pCq, satisfying coherence axioms (pentagon and triangle).
A monoidal category is called rigid if it comes equipped with left and right dual objects -that means, for every X P Ob pCq there exists respectively an object X ˚P Ob pCq with evaluation and coevaluation maps ev X : X ˚b X Ñ ✶ and coev X : ✶ Ñ X b X ˚, as well as an object ˚X P Ob pCq with evaluation and coevaluation maps r ev X : X b ˚X Ñ ✶ and Ć coev X : ✶ Ñ ˚X b X satisfying in both cases the usual conditions.
A ❦-linear abelian category C is locally finite if, for any two objects V, W P Ob pCq, Hom C pV, W q is a finite-dimensional ❦-vector space and every object has a finite filtration by simple objects.
Further, we say C is finite if there are finitely many isomorphism classes of simple objects.A tensor category is a locally finite, rigid, monoidal category such the the tensor product is ❦-linear in each slot and the monoidal unit is a simple object of the category.
At this point, it is useful to introduce the following notion.Given an abelian category C, the Grothendieck group Gr pCq of C is the free abelian group generated by isomorphism classes X i of simple objects in C. If X and Y are objects in C such that Y is simple then we denote as rX : Y s the multiplicity of Y in a Jordan-Hölder series of X.To any object X in C we can canonically associate its class rXs P Gr pCq given by the formula: for all X, Y P Ob pCq, called the braiding, that are compatible with the monoidal structure of the category.This means, the braiding satisfies the so-called hexagon identities for any three objects X, Y, Z P Ob pCq: A ribbon category is a braided tensor category C together with a ribbon twist, i.e., a natural isomorphism θ X : X Ñ X which satisfies In order to define modular tensor categories, we require the notion of non-degeneracy of a braided category.We say that an object X centralises another object A braided finite tensor category C is non-degenerate if the only objects X that centralise all objects of C are of the form X " 1 'n [EGNO15, Section 8.20].Equivalently, C is non-degenerate if and only if it is factorisable, i.e., there is an equivalence of braided monoidal categories ZpCq » C rev ⊠ C, where C rev is C as a tensor category, but with reversed braiding given by the inverse braiding [Shi19].If C is a fusion category (i.e., a semi-simple finite tensor category) with a ribbon structure, then the above notion of non-degeneracy is equivalent to the commonly used condition that the S-matrix is non-singular.Definition 2.5 ([KL01, Shi19]).A braided finite tensor category is modular if it is a non-degenerate ribbon category.

Z `-Rings.
In this subsection we continue following [EGNO15].Denote as Z `the semi-ring of positive integers with zero.
Definition 2.6.Let R be a ring which is free as a Z-module.
(1) A Z `-basis of R is a basis B " tb i u iPI , where I is an indexing set, such that b i b j " (2) A Z `-ring is a ring with a fixed Z `-basis and with an identity 1 which is a non-negative linear combination of the basis elements.If 1 is a basis element, then it is called a unital Z `-ring.
(3) Given a Z `-ring pR, Bq, a Z `-module is an R-module T with a fixed Z-basis M " tm l u lPL such that for any Example 2.7.For C a fusion category with X i the representatives of the isomorphism classes of simple objects, the tensor product on C induces a natural multiplication on Gr pCq defined by the formula: where i, j P I.This multiplication is associative, and thus Gr pCq is a Z `-ring with unit r1s.Gr pCq is called the Grothendieck ring of C.
Let pR, Bq be a Z `-ring, and let i P I 0 Ă I such that b i appears in the decomposition of 1.Then, let τ : R Ñ Z denote the group homomorphism defined by: Definition 2.8.A Z `-ring pR, Bq is called a based ring if there exists an involution pq ˚: where a i P Z, is an anti-involution of the ring R and Proposition 2.9 (Proposition 3.1.6,[EGNO15]).In any based ring, the number c k i,j is invariant under cyclic permutations of i,j,k.
We arrive to one of the main definitions of this paper: Definition 2.10.A fusion ring is a unital, based ring of finite rank.
In order to later introduce the NIM-reps, it is convenient to observe the following property: Proposition 2.11.(Rigidity property) A fusion ring pR, Bq can be equipped with a symmetric bilinear form p´, ´q : R ˆR Ñ Z satisfying pb i b j , b k q " pb j , b i ˚¨b k q.
Proof.Let p´, ´q be the symmetric bilinear form defined by the condition pb i , b j q " δ i,j .Then the property of τ pb i b j q " δ i,j ˚can be reformulated as It is clear that p1, pb i b j qb k q " τ ppb i b j qb k q " c k ij , and that p1, pb i b j qb k q " p1, b i pb j b k qq by associativity.
Thus, we find that where the first equality uses that the induced map is an anti-involution.Relabelling of the indices gives the stated result.□ Notation 2.12.Whenever clear from the context, we may refer to a fusion ring simply as R instead of pR, Bq.
Example 2.13.Let C be a semi-simple rigid monoidal category.Then the Grothendieck ring is a fusion ring, with the simple objects acting as the basis.The involution is then taking the dual of an object and the symmetric bilinear form is pX, Y q " dim ❦ pHom C pX, Y qq, for simple objects X, Y P Ob(C).
The following explicit examples will be studied in detail in later Sections.
Example 2.14 (Group rings).Take a finite group G, and construct the group ring ZG where addition is linear and multiplication is given by the group operation.Then pZG, Gq is a fusion ring with involution g ˚" g ´1.
Example 2.15 (Ising fusion ring).Let B " t1, X, Y u, and R the integer span ZB with addition defined linearly and multiplication given by the fusion rules pZB, Bq is a fusion ring with the self-dual involution X ˚" X, Y ˚" Y.
2.4.NIM-Reps.In this subsection we introduce the main character of this article, following [BPPZ00, BPRZ21, Definition 2.16.Let pR, Bq be a fusion ring.A non-negative integer matrix representation (NIM-rep for short) of pR, Bq is a Z `-module pT, M q that satisfies the following condition; -(Rigidity condition): let T have a symmetric bilinear form p´, ´q : M ˆM Ñ Z defined by pm l , m k q " δ l,k for any l, k P L. Then we must have, for any i P I, l, k P L Remark 2.17.In this definition, unlike Proposition 2.11, rigidity is a condition, not a property.
Example 2.18.A fusion ring can always be considered as a NIM-rep of itself, with the module action simply the ring multiplication.
Remark 2.19.Note that, for t P T, m l P M , the symmetric bilinear form pt, m l q counts the multiplicity of m l in the basis decomposition of t.We then immediately see that pt, tq " 1 if and only if t P M .
Viewed as Z `-modules, it is straightforward to define the direct sum of NIM-reps: given two NIM-reps pT, M q, pT1 , M 1 q over a fusion ring pR, Bq the direct sum of NIM-reps is the R-module T ' T 1 with a distinguished basis M ' M 1 .Other basic notions like sub-NIM-rep are defined in a similar way.Remark 2.21.Suppose we have a NIM-rep pT, M q over a fusion ring pR, Bq that satisfies b i ▷ m j " 0 R for some b i P B, m j P M .The rigidity condition then imposes that m j " 0 R P M .However, the only way 0 R appears in the NIM-rep basis is if t0 R u " M , (i.e this NIM-rep is the trivial NIM-rep).We shall remove this NIM-rep from future considerations.Definition 2.22 ([BD12]).Let pT, M q, pT 1 , M 1 q be two NIM-reps over a fusion ring pR, Bq.A NIM-rep morphism is a function ψ : M Ñ M 1 inducing a Z-linear map between the modules.If ψ is a bijection, and the induced map is an isomorphism of R-modules, then we say that the NIM-reps are equivalent.
Notation 2.23.Unlike the case of a fusion ring, since we will be working mostly with the basis of the NIM-rep, we may refer to a NIM-rep simply as M instead of pT, M q.
We can visually express the data of a NIM-rep in the following way.For a given NIM-rep pT, M q over a fusion ring pR, Bq, a NIM-graph (sometimes also called in the literature 'fusion graphs' ) is constructed with a node for each element of the basis M , and a directed arrow with source m l and target m k , labelled by an element b i P B, for every copy of m k in b i ▷ m l .Every node in a NIM-graph will have a self-loop labelled by the ring identity, which we omit for simplicity.
Example 2.24.If we consider the Ising fusion ring from Example 2.15 as a NIM-rep over itself, the corresponding NIM-graph is given by; -The NIM-graph allows us to visualise irreducibility of the corresponding NIM-rep, as a NIM-rep is irreducible if and only if the NIM-graph is connected.
-In [Gan05,Gan06], NIM-reps are defined equivalently as an assignment of a matrix with nonnegative integer entries to each element in ObpCq, satisfying several compatibility conditions.We will not use this description in the main text but it will be useful in Appendix A.
Definition 2.26.An algebra in C is a triple pA, m, uq, with A P Ob pCq, and m : being morphisms in C, satisfying unitality and associativity constraints: Example 2.27.Let C be a tensor category, then 1 is an algebra.In fact, for any X P ObpCq the object A " X b X ˚has a natural structure of an algebra with unit u " coev X and multiplication m " Definition 2.28.
Just like in abstract algebra, one can construct the related notion of a module over an algebra in the following way.
Definition 2.29.Take A :" pA, m, uq, an algebra in C. A right A-module in C is a pair pM, ρ M q, where M P ObpCq, and Right A-modules in C and their morphisms form a category, which we denote by Mod C ´A.The categories A ´Mod C of left A-modules and A ´Bimod C of A-bimodules in C are defined likewise.
We want to relate these categories of modules to the following notion: Definition 2.30.Let C be a monoidal category.A left module category over C is a category M equipped with an action (or module product) bifunctor b : called the module associativity constraint and In a similar way, one defines a right C-module category.A tensor category is the simplest example: it is a module category over itself.A convenient, less trivial example for us is the following: Proposition 2.31 (Proposition 7.8.10,[EGNO15]).Mod C ´A is a left C-module category.
In fact, given certain conditions one can go one step further.
Lemma 2.32 (Proposition 7.8.30,[EGNO15]).Let A be a separable algebra in a fusion category C. Then the category Mod C ´A of right A-modules in C is also semisimple.
Example 2.33.(Follow-up from Definition 2.13) If we take a semi-simple module category M over C, then the Grothendieck group GrpMq is a NIM-representation of GrpCq, with the isomorphism classes of simple objects acting as the basis.The NIM-rep action of GrpMq on GrpCq is induced from the module category action of M on C.
In particular, if C is a fusion category, we note that by Lemma 2.32 every separable algebra A in C gives rise to a NIM-rep over GrpCq.
We now describe how to detect potential algebra objects from certain NIM-reps.We state the following theorem using results from [EGNO15, Section 7.10]: Theorem 2.34.Let C be a fusion category, M an indecomposable semisimple C-module category, and N P Ob pMq such that rN s generates GrpMq as a based Z `-module over GrpCq.Then there is an equivalence M » Mod C ´A of C-module categories, where A " HompN, N q.
We now translate this to the language of NIM-reps; Lemma 2.35.Suppose we are in the setup of Theorem 2.34.Then the basis element rN s of the NIM-rep GrpMq satisfies the condition that, for all other basis elements rN i s, there exists a basis element rX i s in GrpCqsuch that rX i s ▷ rN s " rN i s.
Proof.This follows straightforwardly from the conditions in Theorem 2.34, as the class rN s generates GrpMq as a Z `-module.This condition restricted to the basis elements gives the result.□ Definition 2.36.We shall call a NIM-rep pT, M q over the fusion ring pR, Bq admissible if there exists an element m 0 P M such that, for every other element m i P M , there exists an element b j P B that satisfies b j ▷ m 0 " m i .
Proposition 2.37.Let pT, M q be an admissible NIM-rep over the fusion ring GrpCq.If it is the underlying NIM-rep of an indecomposable semisimple C-module category, as in the setup of Theorem 2.34, then the decomposition of the algebra A " HompN, N q is given by À iPI a i b i , where a i is the number of self-loops of m 0 labelled by b i in the NIM-graph of pT, M q.
Proof.Using the isomorphism from [EGNO15, Equation 7.21] applied to the algebra A, we have that Hom C pX, Aq -Hom M pX ▷ N, N q.
By Schur's Lemma, if we restrict X to the simple objects of C, then X appears in the decomposition of A if and only if N is in the decomposition of X ▷ N .But by restricting to the NIM-reps picture, and the identification of m 0 with N , we see that this occurs exactly when X labels a self-loop on m 0 .This gives the result.
As every NIM-rep that can be constructed from an irreducible separable algebra is necessarily admissible, we now have a criteria that allows us to capture all possible object structures of these separable algebras.

NIM-Representations
In the following subsections we compute explicitly the NIM-reps of fusion rings associated to relevant examples of families of modular and fusion categories.We also extract from these any algebra objects and compare to existing results.
From now on, we refer to NIM-reps pT, M q simply as M for the sake of clarity.
3.1.Group Rings.Let G be a finite group.In this section, we will focus on classifying all possible NIM-reps over the group fusion rings RpGq :" pZG, Gq described in Example 2.14.
Proposition 3.1.Let M be a NIM-rep over the fusion ring RpGq.The NIM-rep module action restricts to a group action on M .
Proof.The NIM-rep module action will restrict to a group action on M if every element g ▷ m l is in the basis of M .This can be seen as pg ▷ m l , g ▷ m l q " pm l , g ´1 ▷ pg ▷ m l qq " pm l , m l q " 1 and so g ▷ m l is in the basis of M by Remark 2.19.□ If we restrict ourselves to irreducible NIM-reps, we get the following result.
Proposition 3.2.Irreducible NIM-reps of the group fusion ring RpGq correspond to transitive group actions of G.
Proof.If a NIM-rep M over RpGq is not irreducible, then its corresponding group action will always be partitioned into G-orbits by restricting the action to the NIM-reps that sum to give M .Thus the group action is transitive only if the NIM-rep is irreducible.Conversely, if the group action is not transitive, then we can write it as the sum of some finite combination of G-actions ▷ i : G ˆMi Ñ M i .It is easy to see that each M i is a Z-basis for another NIM-rep over RpGq.Hence the NIM-rep is irreducible only if the group action is transitive.□ We can thus explicitly describe the structure of such NIM-reps of RpGq.By Proposition 2.3, the basis elements of an irreducible NIM-rep M are parametrised by the left cosets of H in G, for some subgroup H Ď G i.e if we let tg i u 1ďiď|G:H| be a set of coset representatives, then M " tm g i u 1ďiď|G:H| .
The NIM-rep action is then given by the induced G-action on this set of left cosets, where Y g i P g j H.We shall write M pHq for such a NIM-rep.The Klein-four group has presentation There are 3 isomorphism classes of subgroups in Z 2 ˆZ2 ; ‚ Z 2 ˆZ2 as a subgroup of itself; Then M pZ 2 ˆZ2 q has a single basis element corresponding to the single coset representative e.The NIM-rep graph is given by ‚ Isomorphism class of Z 2 ; There are 3 conjugacy classes of subgroups in this case; H 1 " te, au, H 2 " te, bu, H 3 " te, cu.The NIM-reps M pH 1 q, M pH 2 q, M pH 3 q have two basis elements parameterised by coset representatives te, bu, te, cu, te, au respectively.3.2.Near-Group Fusion Rings.In this subsection, we shall focus on another class of fusion rings that can be formed from a finite group G. Definition 3.6 ( [Ost15,Sie03]).Let G be a finite group and α P Z `.The near-group fusion ring is the fusion ring constructed by taking the integer span of the set G Y tXu, with multiplication of the group elements as the group operation, and with the element X as: The element X is self-dual, i.e X ˚" X.This is a fusion ring, which we shall denote by KpG, αq.
Example 3.7.[EGNO15, Example 4.10.5]The case α " 0 is known as the Tambara-Yamagami fusion ring.Notice that this ring is categorifiable if and only if G is abelian.
The action of KpG, αq on a NIM-rep M consists of the action of the group G and the non-invertible element X.By the results of Section 3.1, we know that the NIM-action of the group component will correspond to some G-action on the NIM-rep basis.However, unlike Section 3.1, we cannot guarantee that this G-action is transitive on the basis M , due to the action of the non-invertible element X.This can be seen in the following example: Example 3.8.(NIM-rep over KpZ 2 , 0q) The Ising fusion ring from Example 2.15 can be viewed as the near-group fusion ring KpZ 2 , 0q.Viewed as a NIM-rep over itself, using the notation from Example 2.24, the NIM-basis can be partition into G-orbits, M " tm 1 , m Y u Y tm X u, with stabiliser groups teu and Z 2 respectively.The two orbits are connected by the action of X, with From this example, we see that the NIM-rep basis M can be partitioned into G-orbits, which are connected to each other by the action of X.We will write this partition as where the i-label counts the p distinct orbits, each defined by a stabiliser subgroup tH i u.The l-label denotes the individual elements in each orbit.Hence, a NIM-rep M over KpG, αq consists of ř p i"1 |G : H i | basis elements, with the NIM-rep action of the group part of KpG, αq having already been covered in Section 3.1.Thus, we now need to focus only on the action of the non-invertible element X. Proposition 3.9.Let M be an irreducible NIM-rep over KpG, αq.For a fixed group orbit label i in the partition of M , we have that Proof.As m i l 1 and m i l 2 are in the same G-orbit, there is a group element g P G such that m i l 1 " g ▷ m i l 2 .Then, using the module action and the fusion rules in Definition 3.6, we have that □ Notation 3.10.We shall write c i,j :" pX ▷ m i l , m j k q.It is clear from Proposition 3.9 that varying the l, k labels has no effect.Additionally, note that by the rigidity condition of the NIM-rep, we have c i,j " c j,i .
Remark 3.11.Irreducibility of a NIM-rep over KpG, αq implies that the group orbits are connected to each other by the non-invertible element X.
From now on, we shall denote the action of X on an element m i l by If we act on both sides of Equation (3.2.1) with X, using the fusion rules in Definition 3.6 along with Proposition 3.9, we find that By counting in this equation the multiplicities of the orbit labelled with i, and one labelled by q ‰ i respectively, we get that Remark 3.12.Should our NIM-rep partition contain only one G-orbit, then we would only have one equation of the form of Equation (3.2.2) as we clearly can pick no q such that q ‰ i.
To classify all NIM-reps over a near-group fusion ring, we thus need to find solutions for this set of equations.To better visualise this problem, we can think of these equations in terms of matrices.By setting C :" tc i,j u, the matrix that determines the action of X, and B "diagt|G : H i |u, the action of X is thus governed by the matrix equation (3.2.4) CBC " α ¨C `|G| ¨B´1 .
We are thus seeking to find choices of subgroups tH i u such that there is an non-negative integer-valued symmetric matrix C satisfying Equation (3.2.4).
As the matrix B is invertible, we obtain a quadratic matrix equation in the variable CB; where I is the identity matrix.As all of the coefficient matrices commute with each other, we can solve via the quadratic equation, giving us where Y is a square root of the identity matrix I.As all the elements in CB are non-negative, the non-diagonal elements of Y must all have the same sign.As both Y and ´Y are square roots of the identity matrix, only one can provide a valid solution for CB, so we can always take the sign in Equation (3.2.5) to be positive.
Remark 3.13.As the elements of CB are integers, the non-diagonal elements of Y must be divisible by p α 2 4 `|G|q ´1{2 .Hence, the problem of classifying NIM-reps over KpG, αq comes down to finding suitable square roots of identity matrices.We do this in full detail for p " 1, 2 in the following propositions.
Proposition 3.14.NIM-reps over KpG, αq consisting of one group orbit are parametrised by pairs pH, c 1,1 q, where H Ď G a subgroup and c 1, Proof.As p " 1, Y " 1 is the only possible choice that leads to a valid solution.Let H be the subgroup of G that governs this orbit.The action of the non-invertible element X is given by a single non-negative integer c 1,1 P Z `.By Remark 2.21, this integer is in fact strictly positive.
If we switch to the element-wise notation, Equation (3.2.5) can be written as By squaring both sides, Rearranging this for α gives the required equation, with the other conditions following from the condition that α must be a non-negative integer.□ Proposition 3.15.All irreducible NIM-reps over KpG, αq consisting of two group orbits are parametrised by tuples pH 1 , H 2 , c 1,1 , c 2,2 q, where H 1 , H 2 Ď G are subgroups and c 1,1 , c 2,2 P Z `, such that α " c 1,1 |G : Proof.It is well known that all square roots Y of the 2-by-2 identity matrix are either diagonal matrices whose non-zero elements are from the set t´1, 1u, or have the form It is clear by Remark 3.11 that irreducible NIM-reps can only come from square root matrices of the form in Equation (3.2.6).
Let H 1 , H 2 be subgroups of G that govern the group orbits in the NIM-rep, and C " tc i,j u 1ďi,jď2 the matrix governing the NIM-action of X.By inputting this data into Equation (3.2.5), we get the following system of equations: By adding the first two equations, we get that α " c 1,1 |G : By multiplying the first two equations together and the last two equations together, we obtain By using the defining relation in Equation (3.2.6), and rearranging to solve for pc 1,2 q 2 , we find that Hence we can always obtain c 1,2 from the other input data.The remaining conditions come from the fact that c 1,2 must be a positive integer.
Hence, a two-orbit NIM-rep over KpG, αq has input data of pH 1 , H 2 , c 1,1 , c 2,2 q satisfying the above conditions, and explicit X-action given by

□
While this only completes the classification for NIM-reps consisting of two group orbits, this is sufficient to completely classify irreducible NIM-reps over the Tambara-Yamagami fusion rings.
Corollary 3.16.All irreducible NIM-reps over the Tambara-Yamagami fusion ring KpG, 0q consist of at most two group orbits.Proof.By setting α " 0 in Equation (3.2.4), we easily see that the matrix CBC must be diagonal.Element-wise, this means that for all i ‰ j.However, by Remark 3.11, there always exist a choice of i ‰ j such that both c i,k , c i,j ‰ 0 when p ě 3. Thus, as C is symmetric, the result follows.□ Example 3.17.(NIM-reps over the Ising fusion ring) Recall Example 2.15.The Ising fusion ring can be viewed as the Tambara-Yamagami fusion ring KpZ 2 , 0q.So by Corollary 3.16, we only need to check for 1 and 2-orbit NIM-reps.
When p " 1, we must have c 1,1 |Z 2 : H| " |H|{c 1,1 by Proposition 3.14.But as the only choices for H are the trivial group and Z 2 , both of which lead to non-integer values for c 1,1 , we see that there are no 1-orbit NIM-reps over KpZ 2 , 0q.
Thus, there is only one irreducible NIM-rep over the Ising fusion ring, given by the tuple pt1u, Z 2 , 0, 0q.
For p ě 3, there is not an explicit classification of square roots of the identity matrix.For instance, it is still unknown whether there exists a Hadamard matrix of order 4k for every positive integer k.So, in what follows we will detail some relevant examples.
Example 3.18.A Hadamard matrix of order n satisfies H n H n " nI n , and rows are mutually orthogonal and the matrix contains only `1 and ´1, so clearly 1 ?n H n is a square root of the identity.However, H n contains exactly npn ´1q{2 elements that are ´1, which contradicts the fact all non-diagonal elements of Y must be non-negative, of which there are npn ´1q.So the only value valid is n " 1, which is the trivial case.
Corollary 3.21.Suppose M is an irreducible NIM-rep over Kpg, αq that can be partitioned into p G-orbits.
c i,q |G : H j | for any orbit labels i ‰ q where c i,q ą 0.
Proof.For the p " 1 case, we simply rearrange Equation (3.2.2) for α.The p " 2 case is just as immediate, instead rearranging Equation (3.2.3).Using the fact c i,1 is non-zero by Remark 3.11, we always have at least one choice of matrix elements to write α in this form.□ We conclude this section with some concrete examples.
Example 3.22.An example of such a NIM-rep is given by the tuple pZ 75 , Z 75 , 5, 5q over the fusion ring KpZ 75 , 10q.As the two subgroups that govern the partition of the NIM-rep are the whole group, orbits consist of one NIM-element each, so M " tm, nu.
By the found formulas, c 1,1 " c 2,2 " 5, c 1,2 " 10 and so the NIM-rep has the following structure; We note that by [Ost15, Theorem A.14], this instance is not a categorifiable one as Z 75 is abelian.However, if we let G " pZ 5 ˆZ5 q ⋊ Z 3 , which has order 75 but is not abelian, then the NIM-rep pG, G, 5, 5q with the same structure as above is a NIM-rep over KpG, 10q.Note that this case can admit a categorification.
Example 3.23.KpZ 175 , 62q has a NIM-rep corresponding to pZ 35 , Z 25 , 11, 1q.Labelling the two group orbits by tm i u 1ďiď5 , tn j u 1ďjď7 , the explicit structure of the non-invertible element is given by Note that by [Ost15, Theorem A.6], this fusion ring is not categorifiable.

Admissible NIM-Reps and Algebra Objects for Pointed and Near-Group Fusion
Categories.Finally, we extract algebra objects from the NIM-reps we have computed.While we cannot read out explicit algebra morphisms, we still are able to recover a collection of familiar results.
Proposition 3.24.(Admissible NIM-reps for group rings) All irreducible NIM-reps over a group fusion rings RpGq are admissible.
Proof.Let M pHq be a NIM-rep over RpGq.Recall that the basis elements are parametrised by the left cosets G{H.For any two NIM-basis elements m g i , m g j , we have that pg j g ´1 i q ▷ m g i " m g j .Hence any NIM-basis element can take the role of m 0 in Lemma 2.35, thus M pHq is admissible.□ Remark 3.25.The algebra object associated to a NIM-rep M pHq over RpGq is then given by À hPH b h .It is clear from Examples 3.4 and 3.5 that this object can be seen by counting the self-loops present in the NIM-graph.This agrees with the classification of algebras in pointed categories given in [Nat17,Ost03b].Proposition 3.26.An admissible irreducible NIM-rep over a near-group fusion ring consists of either one group orbit, parameterised by pH, c 1,1 q, or two group orbits, parametrised by pH 1 , H 2 , 0, c 2,2 q.
Proof.For a NIM-rep consisting of one orbit, we can argue in the same way as Lemma 2.35 that any NIM-basis element can be set as m 0 in Lemma 2.35.
For a NIM-rep consisting of more than one group orbit, the only to have an m 0 that connects to the orbits it does not lie in is via the non-invertible element X.So for n P M , where n is not in the same orbit as m 0 , we must have X ▷ m 0 " n.But then by the fusion rules From this we see that the group orbits of m 0 and n are only connected to each other, so by the irreducibility of the NIM-rep it only contains two group orbits.
If we label the orbit containing m 0 by 1, and the one containing n by 2, we note that the condition X ▷ m 0 " n implies c 1,1 " 0. This gives the result.□ Remark 3.27.We thus have two forms of algebra objects arising in categories associated to near-group fusion rings.
‚ For a NIM-rep paramatrised by pH, c 1,1 q, the algebra is given as an object by ' hPH b h 'c 1,1 X. ‚ For a NIM-rep parametrised by pH 1 , H 2 , 0, c 2,2 q, the algebra object is given by ' hPH 1 b h , i.e it is of the form of a group algebra object.
Algebra objects representing module categories over near-group categories have been studied previously at [MM12,Gal12]: in the case of non-group theoretical Tambara-Yamagami categories with abelian G we have twisted group algebras, see [Gal12,Proposition 5.7].[MM12, Section 8 and 9] proceed in a slightly more general way, for G-graded fusion categories with G not necessarily abelian.Only for the case of Tambara-Yamagami (see [MM12, Section 9]) they recover the same two families of algebras we observe.
3.4.pA 1 , lq 1 2 Fusion Rings and its Admissible NIM-Reps and Algebra Objects.Following [NWZ22] (other useful references are [EP21,FK93]), we can construct a modular tensor category pA 1 , lq from a quantum group of type A 1 at level l P Z `.The Grothendieck ring GrppA 1 , lqq has basis tV i u iPr0,ls , and the fusion coefficients of V i V j " ř l k"0 c k i,j V k are given by: (3.4.1) c k i,j " # 1, if |i ´j| ď k ď minpi `j, 2k ´i ´jq and k " i `j mod 2, 0, else.
We note here that, as seen by the fusion rules, GrppA 1 , lqq is commutative.
Definition 3.28.For an object V i P GrppA 1 , lqq, we define the length of V i to be lengthpV i q :" When l is a positive odd integer, we can define a modular subcategory pA 1 , lq 1 2 by taking the full subcategory with simple objects Remark 3.29.In the fusion ring GrppA 1 , lq 1 2 q, it is simple to check using the fusion rules that V i ‰ V j if and only if lengthpV i q ‰ lengthpV j q.
We shall now focus on finding the admissible NIM-reps of GrppA 1 , lq 1 2 q.As l " 1 results in the trivial ring, we shall assume l ě 3. Recall from Lemma 2.35 that an admissible NIM-rep M over a fusion ring has a distinguished basis element m 0 .We have seen in Section 3.3 that in the case of group and near-group fusion rings, we can have objects in the ring basis b i , b j such that b i ▷ m 0 " b j ▷ m 0 , due to the invertibility of the group parts of these fusion rings.This is not the case when working with GrppA 1 , lq 1 2 q.
Lemma 3.30.Let M be an admissible NIM-rep over GrppA 1 , lq 1 2 q, and Proof.If we assume that V i ▷ m 0 " V j ▷ m 0 , then as the fusion ring is commutative, we have that But by Remark 3.29, lengthpV i q ‰ lengthpV j q, and so the only way that i or V 2 j (whichever has larger length), satisfies V k ▷ m 0 " 0. This has been ruled out by Remark 2.21.□ Remark 3.31.If we have objects V i , V j P GrppA 1 , lq 1 2 q such that lengthpV i q ą lengthpV j q, then it is easily verified using the fusion rules in Equation (3.4.1) that pV 2 i ▷ m p , m p q ą pV 2 j ▷ m p , m p q. Thus, if V i ▷ m p P M , then we immediately have that V j ▷ m p P M .In the case of an admissible NIM-rep M over GrppA 1 , lq 1 2 q, if the basis M has cardinality d, we immediately get that the objects V j that satisfy V j ▷ m 0 P M are exactly those of lengthpV j q ď d.Proposition 3.32.In any NIM-rep M over the fusion ring GrppA 1 , lq 1 2 q, pV l´1 ▷ m p , m q q ď 1 for all m p , m q P M .
Proof.If we assume that pV l´1 ▷ m p , m q q " a q l´1,p ě 2, we can use the fusion rules in Equation (3.4.1) to obtain Applying the form p´, m p q, and using the rigidity condition of the NIM-rep, we find that (3.4.2) pV 2 ▷ m p , m p q ě a q l´1,p pV l´1 ▷ m q , m p q ´1 ě 3 The fusion rules in Equation (3.4.1) give that when l ě 3, V 2j V 2 " V 2j´2 `V2j `V2j`2 , when 1 ď j ď l´3 2 , and V 2 V l´1 " V l´3 `Vl´1 .We let h i,p :" ř kPL a k i,p , which counts the number of NIM-basis elements in the decomposition of V i ▷ m p .Applying the fusion rules to By noting that h i,p ą 1 for all choices of i, p, and a p 2,p ě 3 by Equation (3.4.2), when we count the NIM-basis elements on each side we obtain the following inequalities; where the last inequality in the second line follows due to our initial assumption.However, the last inequality cannot be satisfied as the fraction on the left-hand side is strictly less than 3 for all values of l, which contradicts pV 2 ▷ m p , m p q " a p 2,p ě 3. Hence we have a contradiction, and so, pV l´1 ˚mp , m q q ď 1, for all m p , m q P M .□ Lemma 3.33.In any NIM-rep M over GrppA 1 , lq 1 2 q, pV 2 l´1 ▷ m p , m p q ď 3 for all m p , m q P M .
Proof.If we assume that pV 2 l´1 ▷ m p , m p q ą 4, then by Proposition 3.32, we have that h l´1,p " pV 2 l´1 ▷ m p , m p q ą 4. By the fusion rules in Equation (3.4.1), we have that pV 2 ▷ m p , m p q ě 3. We are in a very similar setup to the proof of Proposition 3.32, which if we follow through results in the inequality 3l ´9 l `1 ě h 2,p .This fraction is also strictly less than 3, so we obtain the desired contradiction.□ Proposition 3.34.Let M be an admissible NIM-rep over GrppA 1 , lq 1 2 q.Then there exists no Proof.Assume there exists some m k P M such that pV 2 k´1 ▷ m k , m k q " 3.By Lemma 3.33, we can write V k´1 ▷ m k " m x `my `mz , where m x , m y , m z P M are distinct NIM-basis elements.The fusion rule of V 2 l´1 gives us that pV 2 ▷ m k , m k q " 2.
As the NIM-rep is admissible and m x , m y , m y are distinct NIM-basis elements, the cardinality of the NIM-basis M is at least 3, so by Remark 3.31 we know that V 2 ▷ m 0 , V l´1 ▷ m 0 P M .We also know that there exists a V j P pA 1 , lq 1 2 such that V j ▷ m 0 " m k .Using the fusion rules, we find that Acting with V l´1 again on both sides and then using the form p´, m k q, we have GrppA 1 , lqq have been classified in [EK9503] and are in one-to-one correspondence with simply laced Dynkin diagrams with Coxeter number h " l `2.In the case that l is an odd integer, this gives the only NIM-rep as GrppA 1 , lqq viewed as a NIM-rep over itself.The restriction of this NIM-rep to GrppA 1 , lq 1 2 q is exactly the single admissible NIM-rep found in Proposition 3.35.
What we see is that for rank 2,3 MTCs each modular invariant has a corresponding NIM-rep, and the correspondence is true.However, at rank 4 it already starts breaking down: the Z 4 MTC has a NIM-rep that does not correspond to a modular invariant, and the toric and pD 4 , 1q MTCs each having two modular invariants associated to a single NIM-rep.

‚‚‚‚
The trivial subgroup H " teu; The basis elements of M pHq are simply parameterised by elements of Z 2 ˆZ2 . 5. (NIM-reps of RpD 3 q) Consider the dihedral group D 3 , with presentationD 3 " tx, a | x 2 " a 3 " e,xa " a ´1xuThere are four conjugacy classes of subgroups of D 3 , giving four isomorphism classes of NIM-reps; ‚ D 3 as a subgroup of itself.This NIM-graph simply consists of a single basis element, with each group ring element acting trivially.The isomorphism class of Z 3 , given by H " te, a, a 2 u.The isomorphism class of Z 2 is given by 3 conjugate subgroups, H " te, xu, te, xau, te, xa 2 u.This gives one NIM-graph, up to isomorphism of NIM-reps.We shall label our graph using the subgroup H " te, xu; The trivial subgroup H " teu.The basis elements are parameterised by the elements of D 3 ; Modular invariants, NIM-reps and their respective exponents for rank 3 modular tensor categories

□
Example 2.38.Consider the Ising fusion ring as NIM-rep over itself.By looking at the NIM-graph from Example 2.24, we see that this is admissible by setting Proposition 3.3.Two NIM-reps M pHq, M pKq over RpGq are isomorphic if and only if H, K are conjugate subgroups of G.

1 1 Table 1 .
Modular invariants, NIM-reps and their respective exponents for rank 2 modular tensor categories