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The structure of normal lattice supercharacter theories

The character theory of finite groups has numerous basic questions that are often already quite involved: enumerating of irreducible characters, their character formulas, point-wise product decompositions, and restriction/induction between groups. A supercharacter theory is a framework for simplifying the character theory of a finite group, while ideally not losing all important information. This paper studies one such theory that straddles the gap between retaining valuable group information while reducing the above fundamental questions to more combinatorial lattice constructions.


Introduction
Through the work of André [3] and Yan [9], supercharacter theory has allowed us to apply the tools of character theory to groups previously deemed intractable. The more general framework developed by Diaconis and Isaacs [6] further fleshes out a theory that can be adapted to different characteristics one might wish to emphasize in groups (e.g. if one wants to study real-valued characters). However, the construction and existence of supercharacter theories remains somewhat mysterious. There are some basic techniques that apply to all groups, and in some surprising cases (e.g. Sp 6 (2) [6]) these give a complete understanding. Nevertheless, in most cases we do not have a good understanding of what supercharacter theories are possible.
In his thesis work, Aliniaeifard [1] developed an alternate approach that centers on the set of normal subgroups. A standard result in character theory is that knowledge of the characters identifies all normal subgroups of the group; these form a lattice under inclusion. It turns out that every supercharacter theory identifies a sublattice of normal subgroups, and this naturally partitions supercharacter theories according to the sublattices they "see." Aliniaeifard identified the unique coarsest supercharacter theory corresponding to each sublattice, and identified numerous desirable characteristics exhibited by this theory.
This point of view naturally leads to a notion of "simple" supercharacter theory, or one which only identifies the trivial subgroup and the whole group (e.g. simple groups only have simple supercharacter theories). Burkett [5] showed that there is a Jordan-Hölder type factorization for supercharacter theories into simple supercharacter theories, and has developed a framework for super versions of various chains of normal subgroups.
This paper examines the notion of a normal lattice supercharacter theory a bit more closely, tying the character theory to the underlying lattice in a more explicit fashion. There are numerous problems in character theory that generally have difficult solutions, such as explicit character formulas for the irreducible characters, decompositions of tensor products (e.g. Kronecker products as in [4]) and restrictions between groups, etc. It turns out that many of these problems have elegant solutions in the case of normal lattice supercharacter theories, and this paper would like to make the case that while these theories are non-trivial, they retain high levels of computability.
We begin with a review of the necessary lattice notation and an introduction to supercharacter theories. We then proceed to Corollary 3.5 which gives an explicit multiplicative character formula for the supercharacters, reminiscent of the combinatorial character formulas found in [9] for the maximal unipotent subgroups of GL n (q). Under some additional hypotheses that guarantee some level of generality, we also give a decomposition of the point-wise product in Corollary 3.8.
In the case that the normal subgroups form a distributive lattice, we can say even more. An advantage of the normal lattice supercharacter theories is that they are somewhat more canonical, analogous to how every group has a partition by conjugacy classes. This feature allows us to better compare supercharacter theories of groups related via inclusion. We conclude with a description of restriction between finite groups H ⊆ G in Theorem 3.15, where we explicitly decompose the restricted supercharacter theory in terms of the supercharacter theory of the subgroup. This result includes some natural compatibility conditions suggested by the underlying theory.
The original motivation for this work comes out of the supercharacter theory of non-nesting partitions for pattern groups [3]. However, to keep this paper in a more manageable form, we have separated the applications to these groups into a companion paper [2], where this point of view allows us full control of a corresponding Hopf algebra, along with the structure constants on the supercharacter basis (something previously was deemed quite difficult in analogous situations). In this paper we instead illustrate the theory with several families of abelian groups including cyclic groups and vector spaces.

Preliminaries
This section will fix some of the standard notation on lattices in Section 2.1, and in Section 2.2 we review both the definition of supercharacter theories and the construction of the normal lattice supercharacter theory as defined by [1].

2.1.
Lattices of normal subgroups. The goal of this section is to fix notation for lattices with a particular emphasis on lattices of normal subgroups of a finite group.
A lattice L is a poset such that for each pair (K, L) ∈ L × L there is a unique least upper bound K ∨ L and a unique greatest lower bound K ∧ L. We say an element L ∈ L covers K ∈ L if L is minimal with the property that K ≺ L L and K = L. Given L ∈ L, let Given a poset P, let J ∨ (P) = {I ⊆ P | i ∈ I, j ∈ P, i ≺ P j implies j ∈ I} be the lattice of co-ideals, ordered by inclusion.
Algebraic Combinatorics, Vol. 3 #5 (2020) A distributive lattice L is a lattice such that for K, L, M ∈ L, An element K ∈ L is meet irreducible if |C(K)| = 1. The fundamental theorem for finite distributive lattices says that L ∼ = J ∨ (M) where M is the subposet of meet irreducible elements of L. This implies that given any K ∈ L there exists a unique antichain A of M such that K = A∈A A.
Our main example of lattices are sublattices of normal subgroups of a finite group G. Specifically, let ker(G) = {N G} be the lattice ordered by inclusion. In this case, given M, N ∈ ker(G), A sublattice N of ker(G) will be a subset such that For a subset A ⊆ ker(G), let Examples 2.1.
Cyclics. Let C n = x be a cyclic group. Then ker(C n ) is isomorphic to the lattice of divisors of n. For example, In general, and ker(C n ) is a distributive lattice with meet irreducible elements {C m | m < n, n/m is a prime power}.
Vector spaces. Let F q be the finite field with q elements, and V an F q -module. Then V is a finite (elementary) abelian group. While the usual character theory of V sees all the normal subgroups of V , we would prefer some sublattices that respect the vector space structure. Thus, and for a fixed basis B ⊆ V , which is isomorphic to the lattice of subsets of B.
The lattice subsp(V ) is not distributive in general, but subsp B (V ) is distributive with meet irreducible elements 2.2. Supercharacter theories. Supercharacter theories give us a framework for simplifying the character theory of a group while maintaining the representation theoretic underpinnings. While one can view them as central Schur rings, this section outlines the more representation theoretic framework introduced by Diaconis-Isaacs [6].
Given a set partition K of G, let be the set of functions constant on the blocks of K.
A supercharacter theory S of a finite group G is a pair (Cl(S), Ch(S)) where Cl(S) is a set partition of G and Ch(S) is a set partition of the irreducible characters Irr(G) of G, such that We refer to the blocks of Cl(S) as the superclasses of S, and the elements of Given a group G with a supercharacter theory S, we call a normal subgroup Snormal if it is a union of superclasses. These normal subgroups can also be defined as intersections of kernels of the corresponding supercharacters [8], and the set ker(S) = {N G | N a union of superclasses} forms a sublattice of all normal subgroups [1]. Note that the trivial supercharacter theory of conjugacy classes gives all normal subgroups ker(G). More generally, given an arbitrary sublattice of normal subgroups of G, one might ask which supercharacter theories (if any) see at least those normal subgroups. The following supercharacter theory defined in [1] gives the coarsest such supercharacter theory for each sublattice of normal subgroups.
(a) The partitions Note that [1] also provides recursive supercharacter formulas for the supercharacters and proves a number of nice properties (e.g. these supercharacters are integer valued). Just as a normal subgroup is a union of superclasses, for each normal subgroup N G there is a natural character that is a "union" of supercharacters with character formula Cyclics. Let C n = x be a cyclic group. Then for d | n, is n/d}. Note that in this supercharacter theory, superclasses and supercharacters are indexed by divisors of n, so for d | n write Note that this supercharacter theory was used (though not defined in this way) to study Ramanujan sums in [7].
Vector spaces. For the lattice subsp(V ), we have that for U ∈ subsp(V ) and In this case, the superclasses and supercharacters are indexed by the number of one and zero dimensional spaces of V or On the other hand, for the lattice subsp B (V ), we have that for A ⊆ B In this case, the superclasses and supercharacters are indexed by subsets of B, so for

Normal lattice theories
This section includes our main results. We begin in Section 3.1 by constructing the dual lattice of ker(S) in terms G-modules. This gives a recursive construction for the modules corresponding to the supercharacters in S. We use the module structure to deduce several degree sum results in Section 3.2, allowing us to also obtain an explicit character formula. In Section 3.3 we find a decomposition for the point-wise product in some situations, and we conclude in Section 3.4 with a decomposition of the restriction of supercharacters under some beneficial assumptions.
3.1. Supermodules. Given a normal subgroup N ⊆ G, we obtain the permutation module From the last isomorphism, we may view each of these modules as submodules of the regular module CG (the permutation module coming from the normal subgroup {1}). Note that by (5), Then by the character decomposition formula (4) of χ M , In particular, taking dimensions, we obtain The next result gives a better understanding of modules that arise in this way. Given a subgroup N ⊆ G, the notation G/N will denote a transversal for the cosets with 1 ∈ G/N . Also, recall (2) for the definition of A.
Lemma 3.1. Let ker(S) be a lattice for a supercharacter theory of G. Fix M ∈ ker(S) and let A be an antichain in the interval Then (a) There exists a choice of transversals N/M and G/A such that Proof. For the proof, order the elements of A = {N 1 , N 2 , . . . , N }, and for each 1 (a) Fix coset representatives G/M with 1 ∈ G/M , and for each 1 j choose coset representatives in such a way that Conversely, for each W j there exists e Nj such that is a basis, and so we get the corresponding dimension formula. 3.2. Character formulas. This section works out some general character formulas that are mostly direct consequences of Lemma 3.1. The first result examines a general character degree sum that carves out a piece of a lattice of normal subgroups ker(S).
We therefore apply Lemma 3.1(a) to We have that C Proof. This is the case where L = G and K = M in Theorem 3.3.
Since this situation satisfies the hypotheses of Lemma 3.1(b), we have an explicit basis and can use this basis to compute the trace of the module.
Algebraic Combinatorics, Vol. 3 #5 (2020) Proof. Suppose g ∈ G, then Suppose g ∈ B with B nonempty and minimal. If N, N ∈ B are not equal, then since they are covers Thus, we get a coefficient if for each N ∈ C(M ) either g N = 1 or g N = a N . In fact, in this case as desired.

Normal lattice supercharacter theories
Examples 3.6. Cyclics. Consider ker(C n ) and fix b | n. Let and for O ⊆ P b , let O = p∈O p. By Corollary 3.5,

In [1, Theorem 3.4], Aliniaeifard gives a lattice theoretic formula for the character values given by
In the examples of ker(C n ) and subsp B (V ) above, the Möbius functions for the divisor lattice and subset lattice are well-known, so we get some identities for free. For a set of distinct primes P b and a | P b , where we cancel the n/b that appears on both sides. In the second example, for D ⊆ B A⊆C⊆B D⊆C

Tensor products.
For any supercharacter theory S of G, the space f(G; Cl(S)) is closed under point-wise multiplication. However, as with irreducible characters it is often not obvious how to decompose the point-wise product of two supercharacters into supercharacters (meanwhile the superclass identifier functions are orthogonal idempotents). However, this problem seems far more tractable for normal lattice supercharacter theories. For M ∈ ker(S) let C(M ) be its set of covers. The following lemma is framed in the language of groups, but applies to arbitrary modular lattices. .
Then Corollary 3.5 implies as desired.
Cyclics. Let C n be cyclic and suppose a, b | n. Fix C p gcd(a,b) ∈ C (C gcd(a,b) ). Then a,b) . In other words, the hypotheses of the Corollary 3.8 are satisfied if Vector spaces. where I are the intersection-irreducible subgroups of ker(S). Thus, for each normal subgroup N ∈ ker(S), there exists a unique antichain A ∈ AntCh(I) such that in the notation of (2), Since the dual of a distributive lattice is distributive, we also have that if P is the set of product irreducible elements, then for each N ∈ ker(S) there exists B ⊆ AntCh(P) such that Proof. (a) Fix L such that B ∈ C(L). Then there exists K ∈ B such that K = K ∩ L. Then KL = B and so K ∈ C(K ∩ L). Suppose there exists L with B ∈ C(L ) such This result allows us to prove some convenient features about distributive lattices of normal subgroups. Proof. (a) For each L such that N ∈ C(L) we can select an element g L ∈ K L −(K L ∩L) so that N ∈C(L) g L ∈ N but N ∈C(L) g L / ∈ L for any L such that N ∈ C(L ). Thus, Since the dual of a distributive lattice is distributive, we obtain the dual to Lemma 3.10 (which seems somewhat harder to prove directly).
Let H ⊆ G be a subgroup and suppose J ∨ (I H ) ⊆ ker(H) and J ∨ (I G ) ⊆ ker(G) are distributive lattices. For the restriction functor Res G H : f (G; Cl(J ∨ (I G ))) −→ f (H; Cl(J ∨ (I H ))) to be well-defined, we minimally require that for each N ∈ J ∨ (I G ), there exists a subset A ⊆ J ∨ (I H ) such that For the restriction result below, we will want slightly stronger compatibility between the lattices. In particular, we say J ∨ (I G ) and J ∨ (I H ) are restriction favorable if (R1) The function Remark 3.13. At first glance it seems that (R2) should always hold due to the diamond isomorphism theorem. In fact, if we use the full lattice of normal subgroups for both groups this is the case. However, (R2) guarantees that the lattice for G is not too coarse with respect to the lattice of H.
Cyclics. For C m ⊆ C n , we have · ∩ C m : ker(C n ) −→ ker(C m ) is well-defined, since ker(C m ) is in fact an interval in ker(C n ). The other condition also follows easily, using (3) and modularity of the lattice.
where all the terms have nonzero coefficients.
Before proving the theorem, we will first need a lemma that computes degree sums using Theorem 3.3.
In this case, by Theorem 3.3, The other terms all satisfy N ∩ A H = K for some N ⊇ (C(A) ∩ H). By Theorem 3.3 and Corollary 3.4, Algebraic Combinatorics, Vol. 3 #5 (2020) and every cover of K sitting in A H appears in this way. Thus, Also, and by distributivity as desired. Therefore we can apply Corollary 3.8 to our situation to obtain Reorganizing, .

K(C(A) ∩ H) ∩ A
. If n = p j1 1 · · · p j and d = p i1 1 · · · p i , then and for i k = j k ,