On residually thin and nilpotent table algebras, fusion rings, and association schemes

Residually thin and nilpotent table algebras, which are abstractions of fusion rings and adjacency algebras of association schemes, are defined and investigated. A formula for the degrees of basis elements in residually thin table algebras is established, which yields an integrality result of Gelaki and Nikshych as an immediate corollary; and it is shown that this formula holds only for such algebras. These theorems for table algebras specialize to new results for association schemes. Bi-anchored thin-central (BTC) chains of closed subsets are used to define nilpotence, in the manner of Hanaki for association schemes. Lower BTC-chains are defined as an abstraction of the lower central series of a finite group. A partial characterization is proved; and a family of examples illustrates that unlike the case for finite groups, there is not necessarily a unique lower BTC-chain for a nilpotent table algebra or association scheme.


Introduction
We explore two related aspects of some important algebraic and combinatorial structures: namely, the properties called residual thinness and nilpotence. The former concept has been studied in the framework of association schemes by Zieschang [13,14] and Hanaki and Shimabukuro [11], among others; and for hypergroups (as an algebraic generalization of schemes) by French and Zieschang [7]. It has been analyzed (evidently independently) in the setting of fusion categories and fusion rings by Gelaki and Nikshych [8]. They use the term "nilpotent" for what the other authors above call "residually thin". This usage has the desirable consequence that a finite group is nilpotent in the classical sense if and only if its representation category (resp. character ring) is nilpotent as a fusion category (resp. fusion ring) [8,Remark 4.7]. However, the group algebra of any finite group, as a fusion ring, is nilpotent according to their definition. This seems to leave room for an alternative definition of nilpotent. The one presented in this paper (see Definition 1.3 below) is a direct generalization of the one given for association schemes by Hanaki [9].
Our context here is the family of table algebras, finite dimensional algebras over the complex numbers with a distinguished basis that satisfies certain axioms (see Definition 1.1 below). The adjacency (or Bose-Mesner) algebras of association schemes, group algebras, and Hecke (double coset) algebras constructed from group algebras are examples, and fusion rings comprise a subfamily. Hypergroups, in the sense of [7], are generalizations of table algebras. Our main results include a formula for the degrees of the basis elements of a residually thin table algebra, and a proof that this formula holds only for such algebras (see Theorem 1.7 below). Gelaki and Nikshych's integrality theorem [8,Corollary 5.3] for fusion categories is an immediate consequence. These are evidently new results for association schemes as well. We also define the notion of thin central chains for a table algebra, in particular upper, lower, and bi-anchored such chains, and use it to exposit our alternate definition of nilpotent table algebra. This definition coincides with the usual one in the case of a finite group; and when applied to commutative table algebras, it is equivalent to both the author's [2] and Gelaki and Nikshych's [8] definitions. Theorems 1.14 and 1.20 below relate residual depth to nilpotence class (see Definitions 1.15, 1.21), and thus extend Gelaki and Nikshych's result for the commutative case [8,Theorem 4.16]. Lower bi-anchored thin-central chains exist for any nilpotent table algebra by Definition 1.11. Theorem 1.23 gives a partial characterization of them. But unlike the case of the lower central series of a finite group, they are not necessarily unique. Example 5.5 shows this, and thereby gives a negative answer to a question of Hanaki [9,Question 2.11] regarding uniqueness of lower central series in an association scheme.
We recall a few well known definitions and facts needed in order to state the main results. These are developed in a number of sources, in particular [1,6,3]. It follows as a consequence of the definition that λ bb * 1 = λ b * b1 > 0 for all b ∈ B. Frobenius-Perron eigenvalue theory yields that for each table algebra there exists a unique algebra homomorphism δ : A → C, called the degree map, such that Let (A, B) be a standard table algebra (STA). For any subsets S, T of B, the set product ST := ∪ s∈S,t∈T Supp B (st), S * := {s * |s ∈ S}, and S + := s∈S s. Note that set product is associative. If T = {t}, a singleton set, then S{t} (resp. {t}S, S{t}S) is denoted St (resp. tS, StS). The order of a subset S is o(S) := δ(S + ).
A nonempty subset C ⊆ B is called closed if C * C ⊆ C. In this case, (CC, C) is again a table algebra, and the set of left cosets Cb for b ∈ B partition B, as do the right cosets bC, and the C-C double cosets CbC. A quotient element, for any b ∈ B, is b//C := o(C) −1 (CbC) + . Let B//C := {b//C|b ∈ B}, and A//C := C(B//C). Then (A//C, B//C) is a STA, called the quotient algebra, or double coset algebra of (A, B) by the closed subset C. Its degree map is δ ↓ A//C , and its anti-automorphism is are in bijection with the closed subsets of B//C via D ↔ D//C (see Proposition 2.5 below). The closed subset C is called normal (resp. strongly normal) in B if bC = Cb (resp. bCb * = C) for all b ∈ B. Strongly normal closed subsets are normal, but the converse is not always true.
An element x ∈ B is called thin (or linear, or grouplike) if xx * = 1. This is equivalent to δ(x) = 1; and if x is thin, then xb ∈ B and bx ∈ B for all b ∈ B. Throughout, Z(A) will denote the center of the algebra A, and Z(B) will mean B ∩ Z(A).

exists a bi-anchored thin chain (BT-chain). It is nilpotent if there exists a bi-anchored thin-central chain (BTCchain).
Remark 1.4. It is immediate from the definition that for any finite group G, the group algebra (CG, G) is nilpotent as a STA if and only if G is nilpotent in the usual group-theoretic sense.
Definition 1.6. Let (A, B) be a STA with a B-chain C: B = C 0 ⊃ C 1 ⊃ · · · C n−1 ⊃ C n = {1}. For any 1 i n and any b ∈ B\C i , define the positive integer m(C, i, b) for i < n by and m(C, n, b) := card(C n bC n ) = card{b} = 1.
We now can state our first main results.
Then the following hold: Theorem 1.7 is proved in Section 3 below, and Theorem 1.8 in Section 4.
i=0 is a TC-chain.
is minimal over all such chains. In particular, a lower BTC-chain of length q is a BTC-chain  The following theorem is proved by French and Zieschang [7, Theorem 6.1] in their more general context of hypergroups. We include a short proof in Section 4, for completeness.

Residually thin and nilpotent table algebras
Definition 1.15. The residual depth of a residually thin STA (A, B) is the minimum length of all BT-chains in B; by Theorem 1.14, it equals r, the length of R.
Remark 1.16. The radical T-chain J in a residually thin STA of depth r may be bi-anchored of arbitrarily large length j > r, or it may not be bi-anchored at all. See Example 4.2 below.
0. An association scheme is called a p-scheme if its adjacency algebra is p-standard. We have the following analog of Theorem 1.14 for thin-central chains and nilpotent STAs. It is a straightforward generalization of [9, Theorem 2.5]. It follows from Theorem 1.14 that for any nilpotent STA, the nilpotence class is at least the residual depth. If A is commutative, then every T-chain is a TC-chain, hence by Theorem 1.20 the residual depth is at least the nilpotence class. So the following result of Gelaki and Nikshych is immediate.

Preliminaries
The results in the section for which proofs are omitted are known; proofs for them are given in [1], [6], or [3]. Throughout, (A, B) is a table algebra (TA).
There is a positive definite Hermitian form ( , ) on A such that for all b, c, d ∈ B, Then ψ preserves the respective anti-automorphisms, ker ψ is a normal closed subset of B, and the following "Fundamental Homomorphism Theorem" holds: Proposition 2.3 has the following consequence:

Residually thin and nilpotent table algebras
Lemma 2.6. Let (A, B) be a STA with closed subsets C 1 ⊇ C 2 and D such that C 2 is strongly normal in C 1 , and for all c ∈ C 1 , cD ⊆ DcC 2 . Then DC 1 and DC 2 are closed subsets such that DC 2 is strongly normal in DC 1 .
Proof. It follows from the hypothesis that Remark 2.7. Given closed subsets C 1 ⊇ C 2 and D with C 2 strongly normal in C 1 , the final hypothesis of Lemma 2.6 will follow from either D normal in B, or both Proof. This is immediate from the definitions if i = n, so assume 1 i < n and b ∈ B\C i . Lemma 2.10 implies that for each j with i j n − 1, Hence, If each C i is normal in B, then each C j //C j+1 is a thin normal closed subset of B//C j+1 , by Proposition 2.5(iv). So Lemma 2.10 yields that for all i j n − 1,

Degrees and double cosets
Throughout this section, (A, B) is a STA with degree map δ, and We prove a theorem that, for a given B-chain, yields a criterion whereby the degree of a quotient element is found in terms of the cardinality of double cosets in quotient bases determined by the chain. The result is applied to prove Theorem 1.7.
Definition 3.1. Fix integer k with 0 < k < n. Define the chain Remark 3.2. We have by Proposition 2.5(i) that Proof. If i = k, then m(C//C k , k, b//C k ) = 1 by definition, as C//C k has length k.
If k = n, then m(C, n, b) = 1, C//C n = C, and b//C n = b. So we may assume that i < k < n. Proposition 2.5(ii) implies for all i j k − 1, so that by Definition 1.6, m(C//C k , i, b//C k ) = i j k−1 card(C j //C j+1 · b//C j+1 · C j //C j+1 ). Since by the same definition m(C, k, b) (resp. m(C, i, b)) equals the analogous product over k j n − 1 (resp. i j n − 1), the result follows.
. Then δ is constant on each double coset C j //C j+1 · b//C j+1 · C j //C j+1 for all 0 i j n − 1 and all b ∈ B\C i if and only if, for all 0 i n − 1, Proof. Fix i 0. By Proposition 2.5(ii), δ is constant on C j //C j+1 ·b//C j+1 ·C j //C j+1 for all i j n − 1 and all b ∈ B\C i if and only if δ is constant on Since B\C i is a union of C n−1 -C n−1 double cosets, Lemma 2.9 implies that (2) is
If C is thin, then C j //C j+1 is a thin closed subset for all 0 j n − 1, by definition. So δ is constant on each double coset for all 0 i n and all b ∈ C i−1 \C i . In particular if i = n, then C n−1 is thin, as noted above. So δ is constant on C n−1 bC n−1 for all b ∈ B.
If i < n and b ∈ C i−1 \C i , then C n−1 thin, Lemma 2.9 and Lemma 3.3 imply that .
So the same hypothesis holds for the chain C//C n−1 in B//C n−1 as for C. Induction on n implies that C//C n−1 is thin. Since C j //C j+1 = (C j //C n−1 )//(C j+1 //C n−1 ) for all 0 j n − 2 by Proposition 2.5(ii), C is thin.

Residually thin STAs
In this section, we prove Theorem 1.14, Theorem 1.8, Proposition 1.18, and in Theorem 4.1 properties of residually thin STAs. Parts of the latter result are proved for hypergroups in [7], as noted below; but we include proofs here, as they fit easily into our context. Example 4.2 below illustrates how much residual depth and radical length can differ. Throughout, (A, B) denotes a STA.
Proof of Theorem 1.14.
, then R n = {1} and n r. The theorem follows.
Proof of Theorem 1.8. Let b ∈ C i . We may assume that b = 1. So for some j with i j < n, b ∈ C j \C j+1 . Theorem 1.7 then implies that o(C j+1 )/δ(b) = m(C, j + 1, b), which is an integer. Lemma 2.11 yields further that By Theorem 1.14, the residual thin chain R is also bi-anchored.   Let (A, B) be a residually thin STA of depth n, with a BT-chain

Residually thin and nilpotent table algebras
D is residually thin of depth at most n, and o(D) is an integer by Proposition 1.5. Thus, (i) follows.
Assume the hypotheses of (ii). Then each DC i is closed, and the hypotheses of Lemma 2.6 hold for each pair C i , C i+1 , by Remark 2.7. By Lemma 2.6, each DC i+1 is strongly normal in DC i . So DC i //DC i+1 = (DC i //D)//(DC i+1 //D) is thin. Therefore, the distinct members of {DC i //D} n i=0 form a BT-chain for B//D, and (ii) is proved. Now C 1 is residually thin of depth n − 1, and is strongly normal in B; and D ∩ C 1 is a closed subset of

4]. That is, H is replaced inside
is defined as follows: for all h ∈ H, y ∈ Y, g ∈ G\H, (hy) · g = hg (the product in G), and g · (hy) = gh. If g 1 , g 2 ∈ G\H with g 1 g 2 = h ∈ H, then in C(C • ψ G), Suppose that H = N G (H); that is, H is its own normalizer in the group G. Then for all b = o(Y )g ∈ B\C, the set product then C is not strongly normal in D, hence D//C is not thin. It follows that the radical thin chain J is {1} ⊂ C, of length 1 and not bi-anchored.
Let n > 1 be an arbitrary integer, and suppose that G is a group with a chain of subgroups (This is the case, for example, if G = D 2 n , the dihedral group of order 2 n , u is a noncentral involution in G, v is the cyclic subgroup of order 2 n−1 , and is the radical thin chain J of B, and it is bi-anchored with length n. Remark 4.3. Regard the groups G, C above as association schemes in the usual way. (For g ∈ G, the relation g L on underlying set G is given by (x, y) ∈ g L iff xy −1 = g Algebraic Combinatorics, Vol. 5 #1 (2022) for all x, y ∈ G; and similarly for C.) The adjacency algebras are isomorphic as table algebras to the group algebras CG and CC, and the partial wreath product constructed above is realized as the adjacency algebra of the wedge product of the schemes [12,Section 3]. So Example 4.2 applies to association schemes.

Nilpotent STAs
We turn now to thin-central chains and nilpotent STAs. Theorems 1.20, 5.2, one direction of Theorem 5.3, and their proofs given below, are generalized directly from Hanaki's results in [9]. Again, (A, B) is always a STA.
Proof of Theorem 1.20. Let {C i } n i=0 , {1} ⊂ C 1 ⊂ · · · ⊂ C n−1 ⊂ C n be a TC-chain. Note first that C 1 is normal in B; and then by Proposition 2.5(v) all C i are normal in B. Now Suppose that C i ⊆ Z i for some i 1. Now C i is strongly normal in C i+1 , and Z i is normal in B. Hence, Remark 2.7 and Lemma 2.6 imply that implies that it is strongly normal. Hence, Q is strongly normal in B by Proposition 2.5(iii). Each C i //C i+1 is normal in B//C i+1 , since C i //C i+1 ⊆ Z(B//C i+1 ) by the definition of a TC-chain. Since C q is normal in B, it follows from Proposition 2.5(v) that each C i is normal in B.
If x ∈ Q i and b ∈ B, then x//Q i+1 thin implies that x//Q i+1 · b//Q i+1 · x * //Q i+1 is a basis element in B//Q i+1 . Since Q is strongly normal in B and x * ∈ Q,

Residually thin and nilpotent table algebras
that the length is at least u; so we have that all terms are distinct and form a terminal BTC-chain of length u, . But minimality of the latter implies that the sums are equal.
Since y//DZ i−1 is thin, both y//DZ i−1 · b//DZ i−1 and b//DZ i−1 · y//DZ i−1 are single basis elements in B//DZ i−1 . So it follows from (4) that for each 1 i m. Thus, Assume furthermore that G u−2 is abelian, and |G u−1 | = p. (This holds, for example, if G is the multiplicative group of upper unitriangular (u + 1) × (u + 1) (u 3) matrices over the field F p .) Let H = G u−2 ; Y = y , a group of order p; C = H × Y , and ψ : C → H be the group homomorphism hy s → h for all h ∈ H, y s ∈ Y . As in Example 4.2, let B be the standard rescaling of the partial wreath product C • ψ G, and A = CB. Then B = {b g := pg | g ∈ G\H} ∪ C, where b g (hy s ) = b g h, (hy s )b g = hb g for all g ∈ G\H, h ∈ H, y s ∈ Y ; and if g 1 , g 2 ∈ G\H, then In particular, y s b g = b g y s = b g , and b g b g −1 = pY + , for all g ∈ G\H. Thus, b * g = b g −1 . Now B//Y ∼ = G via the correspondence b g //Y ↔ g for all g ∈ G\H and hy s //Y ↔ h for all h ∈ H, y s ∈ Y . We so identify the two groups. Define is thin and is central in Because {G i } is the lower central series for G, each Q i , for 1 i u − 2, is the unique minimal closed subset in Q i−1 among all closed subsets that contain Y such that (5) holds. But any closed subset of B that is not contained in C contains b g for some g ∈ G\H, and so contains Supp(b g b * g ) = Y . Therefore, for 1 i u − 2, Q i is in fact the unique minimal closed subset in Q i−1 so that (5) is true. Hence, uniquely minimal for all TC-chains of length u − 2 that proceed down from B.
Let G u−1 = z , of order p by our choice of G. Fix any integer j with 1 j < p, and let D j := z j y . Then G u−1 central in G and Y central in B imply that each D j ⊆ Z(B). Since C is an abelian group (as H = G u−2 is abelian, again by choice Algebraic Combinatorics, Vol. 5 #1 (2022) of G), Q u−2 //D j = C//D j is thin and abelian. Furthermore, for all x = hy r ∈ Q u−2 and b g ∈ B\Q u−2 , we have in the nilpotent group G that hg = ghz t for some integer t. So in B, b g = pg, y r b g = b g y r = b g , and C abelian yield xb g = hy r b g = hb g = b g hz t = b g hy r z t = b g xz t = b g z t x.
Now z t = z js for some integer s, and b g z js = b g (z j y) s . Hence, Therefore, x//D j · b g //D j = b g //D j · x//D j , thus Q u−2 //D j ⊆ Z(B//D j ). Since D j ⊆ Z(B) and o(D j ) = p, it follows that for each 1 j < p, is a lower BTC-chain for B. Each Q i is the unique minimal closed subset of Q i−1 that exists in a BTC-chain, for all i u − 2; but each D j is minimal such in Q u−2 . Thus there are p − 1 stringently minimal such chains, not a unique one. As in Example 4.2, the algebraic construction here is realized as the adjacency algebra of the wedge product of association schemes. Hence this example too applies to association schemes. So the answer to [9, Question 2.11] is negative.
Our final result displays the role played in general by the subsets Supp(b * b) for b ∈ B in finding TC-chains in B from the top down, with each term minimal in the previous one. Proof. If Y is any closed subset with V ⊆ Y ⊆ U , then U//V an abelian group implies that Y //V is a normal abelian subgroup of U//V . Hence by V normal in U and Proposition 2.5(v), Y is normal in U ; and U//Y is an abelian group, in particular is thin. Suppose that C is a closed subset of U such that V ⊆ C, C//V is normal in B//V , and U//C ⊆ Z(B//C). Then U//C thin implies that for any x ∈ U and b ∈ B, x//C · b//C · x * //C = b//C. Hence, (x//V )//(C//V ) · (b//V )//(C//V ) · (x * //V )(C//V ) = (b//V )//(C//V ), which, since C//V is thin and normal in B//V , implies that Suppose that D is another closed subset of U with V ⊆ D, D//V normal in B//V , and U//D ⊆ Z(B//D). Since V ⊆ C ∩ D ⊆ U , U//C ∩ D is also thin. An argument similar to the one above yields that for all x ∈ U and b ∈ B,  We have shown that C ∩ D satisfies the same hypotheses as D, provided that C ⊇ S. Both claims of the theorem follow immediately.