The combinatorics of normal subgroups in the unipotent upper triangular group

Describing the conjugacy classes of the unipotent upper triangular groups $\mathrm{UT}_{n}(\mathbb{F}_{q})$ uniformly (for all or many values of $n$ and $q$) is a nearly impossible task. This paper takes on the related problem of describing the normal subgroups of $\mathrm{UT}_{n}(\mathbb{F}_{q})$. For $q$ a prime, a bijection will be established between these subgroups and pairs of combinatorial objects with labels from $\mathbb{F}_{q}^{\times}$. Each pair comprises a loopless binary matroid and a tight splice, an apparently new kind of combinatorial object which interpolates between nonnesting partitions and shortened polyominoes. For arbitrary $q$, the same approach describes a natural subset of normal subgroups: those which correspond to the ideals of the Lie algebra $\mathfrak{ut}_{n}(\mathbb{F}_{q})$ under an approximation of the exponential map.


Introduction
The unipotent upper triangular group UT n (F q ) consists of all upper triangular matrices with each diagonal entry equal to 1 over the finite field F q with q elements. Uniformly indexing the conjugacy classes or irreducible characters of UT n (F q ) (for every n and q) is an impossibly difficult problem [13]. However, developments in unipotent combinatorics [1,2,5,29] suggest that the difficulty of the conjugacy problem belies nicer combinatorial structures present in the more computable, conjugacy-adjacent properties of UT n (F q ).
Absent a good understanding of the fundamental representation theoretic structures of UT n (F q ), two questions arise: what methods can access the important properties of UT n (F q ), and is there a nice, combinatorial description for the output of these methods? Answers to the former are often algebraic, like the Kirillov orbit method [17,26], or recursive, like the character-theoretic techniques in [15,16,23]. Answers to the latter-when positive-are bijections with families of combinatorial objects; examples include supercharacter theories with classes and characters indexed by labeled set partitions [1] or nonnesting set partitions [2,5], and an indexing of certain irreducible characters with labeled lattice paths [24].
The aim of this paper is to determine whether the normal subgroups of UT n (F q ) belong to the latter universe of bijections and combinatorial objects. Normal subgroups are closely related to conjugacy classes, so it is not clear that the normal subgroups of UT n (F q ) should have a uniform indexing set, and even if such a set exists, a nice exposition remains nontrivial.
What is known about the normal subgroups of UT n (F q ) follows the classical intuition. For a Lie group G, the exponential map exp gives a bijection between closed normal subgroups of G and ideals of the Lie algebra Lie(G). Over F q , the map exp is not well-defined and the right notion of closure is unclear, but morally there should be similar correspondences for nice enough groups, as is the case in the Lazard correspondence [18], and for the finite groups of Lie type. In [20] (see also [19]), Levčuk gives a bijection of this sort for UT n (F q ), using the map a → 1 + a from the nilpotent Lie algebra ut n (F q ) to UT n (F q ) in place of exp.
Theorem A ([20, Theorem 1]). A subset N ⊆ UT n (F q ) is a normal subgroup if and only if N = 1 + n for an additive subgroup n ≤ ut n (F q ) with {[a, b] | a ∈ ut n (F q ), b ∈ n} ⊆ n, where [a, b] = ab − ba is the Lie bracket of ut n (F q ).
Lie algebra ideals are F q -subspaces, so if q is not prime UT n (F q ) has normal subgroups which do not correspond to Lie algebra ideals. (In fact, these are Lie ring ideals.) Let I n (q) = {1 + n | n is an ideal of ut n (F q )} ⊆ {N UT n (F q )}.
The main result of this paper concerns this set of "closed" normal subgroups. When q is prime, this accounts for every normal subgroup of UT n (F q ).
An explanation of the objects in this bijection follows. A tight splice is an apparently unknown object formed by connecting the blocks of a nonnesting set partition according to certain rules. The result is a graph in which each connected component has the shape of a shortened polyomino, a combinatorial object used in [7,8,9] to study Catalan statistics; these statistics appear in my formula for |I n (q)|, see Corollary 4.2. This grid-like shape allows for a labelling scheme in which labels are placed in each "box" of a splice. For example illustrates the labeling of a tight splice with elements of F × 5 (shown in red). This tight splice originates from the nonnesting set partition with blocks {124|356|78} and has the shape of the shortened polyomino (EESS, SESE).
The other object in the bijection is a loopless binary matroid with a labeling. The matroid is given as a bipartite graph (sometimes called a Stanley graph, see Section 2.3 and [27, A135922]) on the set of rows and columns of the tight splice (above, there are two rows and two columns). The labeling assigns an element of F × q to each edge in this graph. Remark. There is a unique F × 2 -labeling for each tight splice and each loopless binary matroid, so in effect there is a bijection with unlabeled objects: S is a tight splice on {1, 2, . . . , n} and M is a loopless binary matroid on the rows and columns of S. .
The proof of Theorem B constructs each ideal of ut n (F q ) from a unique tuple (S, σ, M, τ ). Each labeled tight splice (S, σ) determines a family of ideals via certain shared properties. Selecting an ideal from this family is a matter of F q -linear algebra, which can be encoded into a labeled loopless binary matroid (M, τ ). Considering the set of all normal subgroups in UT n (F q ) when q is not prime, there is no equivalent encoding. Accordingly, I am able to give an outline of sorts for the normal subgroups of UT n (F q ) in terms of tight splices (Corollary 5.7), but not a nice bijective description.
A point of comparison is Marberg's work in [22] on "supernormal" subgroups of UT n (F q ), which relate to two-sided ideals in the associative algebra ut n (F q ) in the same way that ordinary normal subgroups relate to Lie algebra ideals. For prime q, there is a bijection between supernormal subgroups and pairs comprising a nonnesting set partition and a certain type of F q -vector space. Each nonnesting set partition has a trivial tight splice, and in Theorem B, Marberg's subgroups are given by a trivial tight splice (which has a unique labeling) and an arbitrary labeled loopless binary matroid. Thus tight splices are a heuristic for the difference in ideal structure between ut n (F q ) as a Lie algebra and ut n (F q ) as an associative algebra.
The paper is organized as follows. Section 2 is a review of preliminary material, including nonnesting set partitions and matroids. Section 3 is an introduction to tight splices. In Section 4, I construct the ideals of ut n (F q ) from tight splices and loopless binary matroids, proving Theorem B. Finally, Section 5 concerns the normal subgroups of UT n (F q ), giving a formula for the total number of normal subgroups in UT n (F q ) (Theorem 5.9) and several other results; a proof of Theorem A is also included for completeness.
My results suggest a few new lines of inquiry. First, there is the question of normal subgroups in maximal unipotent subgroups for other Lie types; I suspect that my approach will adapt well to the classical cases. Second, normal subgroups form a sublattice in the lattice of subgroups of UT n (F q ). The structure of this sublattice is mostly opaque, but Section 5.2.1 describes the join-irreducible elements of the sublattice. Finally, the connection between tight splices and shortened polyominoes remains somewhat mysterious.

Nonnesting set partitions
] which is an antichain in both L and R . This definition differs from the usual notion of a partition of the set [n]-a collection of disjoint subsets whose union is [n]-but is equivalent: from an antichain λ ⊆ [[n]], the graph on [n] with edge set λ has connected components which divide [n] into pairwise disjoint sets. These connected components also determine λ uniquely: since λ must be an antichian, there edges will only occur between sequential elements of the same connected component.
be the upper set of elements at least covering relations in above some (i, j) ∈ λ. An example of this construction can be seen in Figure 1.
Write ↑ (λ) = ↑ 0 (λ) for the upper set generated by the set λ. The fundamental theorem of finite distributive lattices states that there is a bijection where min(F) denotes the set of -minimal elements in F.

Remark. The number of upper sets of [[n]
] is the nth Catalan number 1 n+1 2n n . One proof (of many, see [28,Problem 178]) is as follows: draw upper sets as in Figure 1 and then rotate counterclockwise by forty-five degrees; the result is a Ferrer's shape which fits inside the shape (n − 1, n − 2, . . . , 1). For λ ∈ NNSP n , each (i, j) ∈ λ corresponds to a removable corner of the shape associated to ↑ (λ).

Normal subgroups and ideals from nonnesting set partitions
The set [[n]] indexes the above-diagonal entries of an n × n matrix, so where δ is the Kronecker delta. The support of a subset n ⊆ ut n (F q ) or 1 + n ⊆ UT n (F q ) is It is straightforward to see that this set will be an upper set in [ , and x ∈ F q . For ideals n, m ⊆ ut n , supp(n + m) = supp(n) ∪ supp(m), so for each upper set F there is a maximal ideal having support F. Under ↑ the set NNSP n indexes these ideals: let By I have drawn a line below the entries corresponding to elements of ↑ (λ) to emphasize the connection between this upper set and UT λ .

Consider the nonnesting set partitions
The corresponding normal pattern subgroups are where Z(UT n (F q )) is the center of UT n (F q ).
The sets ↑ (λ) have an interpretation in terms of ideals and normal pattern subgroups: where [·, ·] first denotes the Lie bracket, and then the group commutator.

Loopless binary matroids
The construction of ideals of ut n (F q ) in Section 4 involves indexing certain F q -vector spaces. If q = 2 this can be done with a certain class of matroid; more generally there is a q-analogue for this class, defined below. This approach is due to the preprint [14] which is not widely available; for the sake of completeness all statements from this source will be proved. A matroid is a pair (K, B) consisting of a ground set K and a collection B of subsets of K called bases, satisfying A loop of (K, B) is an element of K which is not contained in any basis of (K, B). The matroid (K, B) is loopless if it has no loops.
A function φ : K → F r q , r ≥ 0 is an F q -representation of (K, B) if each A ⊆ K satisfies: A ∈ B if and only if φ(A) is an F q -basis for F q -span(φ(K)).
A matroid (K, B) which has an F 2 -representation is said to be binary.
. Let (K, B) be a binary matroid with a totally ordered ground set K, and take A ∈ B to be lexicographically minimal. Then (K, B) is entirely determined by Also, a < b for each a ∈ E b .
As stated at the beginning of the section, a proof will be included for completeness.
Proof. Let φ : K → F r 2 be a representation of (K, B). By assumption φ(A) is a basis for F q -span(φ(K)), so for each b / ∈ A In fact, any pair (A; {E b | b / ∈ A}) satisfying a < b for each a ∈ E b uniquely determines a binary matroid via the F 2 -representation in equation (2.1).

A graph model for loopless binary matroids
A construction of [14] produces a unique graph from each loopless binary matroid; labeling the edges of these graphs will give the "labeled loopless binary matroids" of the main result. This section gives an exposition of this construction, including proofs for completeness. A similar construction for arbitrary matroids is also know, see [25,Section 6.4].
Now, suppose that K is a totally ordered set, and let , and 1 2 3 .

Remark. When
] under the name "Stanley graphs," though this name does not appear elsewhere in the literature.
are mutually inverse, giving a bijection.
shows the six members of → G K . Under Proposition 2.5, the basis of each corresponding loopless binary matroid is, respectively: In light of Proposition 2.5, define the set of F × q -labeled loopless binary matroids on K to be Draw elements of → G K (q) as edge-labeled graphs. The q-Stirling numbers of the second kind are defined by Note that subsets of [k] which contain 1 contribute nothing to the sum; also note that k j q is a polynomial expression in q − 1 with positive coefficients, as Remark. Formula (2.2) is due to [12] (see [6,Equation 10.1] for a recent statement) and is a specialization of the symmetric function h k−j of [21, Section I.2]. Other definitions, including a Stirling-like recurrence, can be found in [6] and [30], among other sources.
. For a totally ordered set K with k elements, Proof. Without loss of generality, take K = {1 < 2 < · · · < k}. It is sufficient to show that Compute → G K (q) by summing over choices of V ⊆ K, which must not contain 1. Fixing V and taking U = K − V , each choice of E and τ is equivalent to a functionτ : Finally,τ is determined by these restricted functions, each of which may be chosen independently.

Splices
It is not obvious, but for a given ideal n ⊆ ut n (F q ) there are relatively few i j ∈ supp(n) for which F q e i,j ⊆ n. These edges occur according to certain straightforward rules, which are formalized in the definition of a tight splice of a nonnesting set partition. Studying a generalization of these rules first leads to a few interesting connections. Let λ and ν be disjoint set partitions of [n]. The set S = λ ν is a splice of λ if: (S1) for each Every set partition λ has at least one splice, the trivial splice λ = λ ∅. Say that λ is the underlying partition of the splice S = λ ν, and that ν is the set of vertical edges of S. Both sets are determined by S: ν = S − λ, and In Sections 4 and 5, λ will always be nonnesting. In this case, the above equation simplifies to λ = min(S), the -minimal elements of S. When drawing a splice S = λ ν, the convention will be to draw edges from λ horizontally and ν vertically, as in (S1) and (S2). For example, Remarks. (R1) In his groundbreaking work on the representation theory of UT n (F q ) [3,4], André defines a set of "superclasses" which are unions of conjugacy classes. Later work [1] indexes these classes with labeled set partitions of [n].
In the language of splices, [4, Theorem 2.2] states that a superclass indexed by the set partition λ is a single conjugacy class if and only if λ has no nontrivial splices. This suggests that splices may be useful in further study of André's superclasses.
(R2) If S is a splice of a nonnesting set partition, each connected component in the graph of a S can be drawn in a grid-like shape, as in (3.1). This shape coincides with that of a shortened polyomino [7,8,9], defined as a pair of lattice paths in Z 2 subject to certain conditions. For example, the splice in (3.1) corresponds to the shortened polyominoes = (ESES, SSEE) and = (EE, EE).
The number of shortened polyominoes with k steps in each path is the kth Catalan number. This connection has yet to be thoroughly investigated, but it does seem to be significant: each important statistic on splices matches a known statistic for shortened polyominoes, and thus other Catalan objects.

Bindings, rows, and columns
Given a splice S = λ ν, define the bindings of S to be elements of so that bind(S) records the tuples of edges to which (S2) applies. If S = λ ν is a splice, then ν gives an equivalence relation ∼ cols on λ, generated by Define the columns of S to be the equivalence classes of λ under ∼ cols , and write including the noteworthy special case of {(i, j)} ∈ cols(S) when (i, j) is not contained in any bindings of S. As a result, Similarly, λ gives an equivalence relation ∼ rows on ν, generated by Let the rows of S be the equivalence classes of ν under ∼ rows , and write This gives a row counterpart to equation (3.2), Unlike columns, (S1) and (S2) ensure that every row has at least two elements. Lemma 3.1. Let S = λ ν be a splice and R a row of S. Let I and K be the connected components in the graph of λ which contain {i | (i, k) ∈ R} and {k | (i, k) ∈ R}, respectively. Then I = K, and if λ is nonnesting, every (i, k) ∈ ν with i ∈ I or k ∈ K belongs to R.
. By (S1) and (S2) S has a binding and either s = max(I), or (s, v) ∈ λ with v > u. In either case, u / ∈ I, so I = K. If λ nonnesting, only s = max(I) can be true because (t, u) ≺ (s, v). Thus if (i, k) ∈ ν with i ∈ I, repeated application of (S2) shows that (i, k) ∼ rows (s, w) for some (s, w) ∈ ν. As ν is a set partition, w = u and (i, k) ∈ R. A similar line of reasoning applied to min({k | (i, k) ∈ R}) shows that (i, k) ∈ ν with k ∈ K also shows that (i, k) ∈ R.
With the description of rows given above, Lemma 3.1 can be used to show that the graph of a splice of a nonnesting set partition is planar and has the properties described in (R2). However, these facts will not be used in the scope of this paper.

Tight splices
Take λ ∈ NNSP n , so that λ is a nonnesting set partition. Say that a splice S of λ is tight if S∩ ↑ 2 (λ) = ∅. By (S2), this is equivalent to the more straightforward condition Proposition 3.2. Let S = λ ν be a tight splice. If λ is nonnesting, then so is ν.

Labeling tight splices
For S ∈ T n , an F × q -labeling of S will be a function σ : bind(S) → F × q . Let Such a labeling can be realized graphically by drawing label values in the center of each binding. For example, with p > 3,

Ordering rows and columns
Let S be a splice and define CR(S) = cols(S) rows(S). Proof. This follows directly from equations (3.2) and (3.3).
The main result of Section 4 requires a uniform total order on the sets CR(S), so as to define F × q -labeled loopless binary matroids on CR(S). One such order is described below. To start, draw S in the usual way, and then lower each successive connected component so that no row of S is directly to the right of another. For example, taking S as in (3.4), we move the second connected component down by one unit: , 9 10 , , and R 2 = .
Finally, enumerate the set CR(S) by first listing columns according to number, and then rows in the same way. This enumeration defines the final ordering, as in: Remark. Aside from ease of description, I do not know of any benefit of this order over any other, at least in the scope of this paper. However, the analogous choice of order for shortened polyominoes is fairly significant (see [8,Section 5]), so it would be interesting to know of a property of ideals or normal subgroups which prefers one order over another.

Lie ideals
Fix a prime power q. This section describes a bijective indexing for the ideals of the Lie algebra ut n = ut n (F q ). Recall the definitions of T n (q) from Section 3.3, CR(S) from Section 3.4, and → G CR(S) (q) from Section 2.3.1.  For each (S, σ), there are ideals D S,σ and Z S,σ such that D S,σ ⊆ n ⊆ Z S,σ for each n ∈ fam(S, σ). Every n ∈ fam(S, σ) is then determined by its image in Z S,σ /D S,σ .
Each F × q -labeled loopless binary matroid (M, τ ) on CR(S) uniquely determines a subspace of Z S,σ /D S,σ which lifts to an ideal in fam(S, σ).

Splices and families of ideals
• a ∈ ut n with a j,k = 0.
Lemma 4.5 should be seen more as a computational shortcut than a precise description of the set [F q e j,j+1 , a] + ut ↑ 2 (λ) : in the event that one or both of a j+1,l or a i,j is zero, several of the cases above are equal.
The lemma is a slightly stronger statement than Proposition 4.3 which will also apply to normal subgroups, as discussed in Section 5.
Lemma 4.6. For λ ∈ NNSP n , let n be an arbitrary subset of ut λ , and suppose that m is an additively closed subset of ut n which satisfies Let ν = {(r, s) ∈ ↑ 1 (λ) | F q e r,s ⊆ m}. Then S = λ ν is a tight splice of λ, and there is a unique labeling σ : bind(S) → F × q for which n ⊆ Z S,σ . Proof of Lemma 4.6. The first step of the proof is to show that S is a tight splice. From the assumption that ut ↑ 2 (λ) ⊆ m it follows that ν ⊆ ↑ 1 (λ)−↑ 2 (λ), so S satisfies conditions (S1) and (T): if r s ∈ ν then r+1 s or r s−1 ∈ λ.
What remains in this step is to show that S satisfies (S2). For each (i, j) ∈ λ we can choose an a ∈ n with a i,j = 0. Using Lemma 4.5 to assist with computation, m contains If (i, j + 1) ∈ ν then in order to avoid a contradiction there must be some l ∈ [n] with (j + 1, l) ∈ λ, a j,l = 0, and F q e j,l / ∈ m. As a consequence, (j, l) ∈ ν, establishing The "if" direction of (S2) follows similarly, establishing that S is a tight splice of λ. The second and final step of the proof is to define a labeling σ of S for which n ⊆ Z S,σ . Fix a binding Using Lemma 4.5 again, for each a ∈ n Since m is closed under addition and F q e i,j+1 , F q e j,l ⊆ m, there must be a single value α i,j,j+1,l ∈ F × q for which a i,j e i,j+1 − a j+1,l e j,l ∈ F q (e i,j+1 − α i,j,j+1,k e j,l ) for every a ∈ n. Such a value exists for each binding of S, giving a labeling σ of bind(S) with Proof of Proposition 4.3. Suppose that n is an ideal with support ↑ (λ) for λ ∈ NNSP n , and let m = n ∩ ut n , so that m is an ideal, and {(r, s) ∈ ↑ 1 (λ) | F q e r,s ⊆ m} = {(r, s) ∈ ↑ 1 (λ) | F q e r,s ⊆ n}.
The converse is almost never true. If S = ∅, there are ideals between D S,σ and Z S,σ which do not belong to fam(S, σ); one example is ut ↑ 1 (λ) , where λ is the underlying partition of S.
Since n ∈ fam(S, σ), it must be the case that F q e i,j+1 , F q e j,l ⊆ n and so that b C ∈ Z S,σ , supp({b C }) = C, and (b C ) i,j = 1 for the unique (i, j) ∈ C with i minimal. Graphically, for each (i, j) ∈ C the (i, j)-entry of b C is the product of the labels above the edge (i, j) in the column C, and all other entries of b C are zero.
Refer to Example 4.7 to verify that these elements, modulo D S,σ , give a basis of Z S,σ /D S,σ .
Lemma 4.11. For (S, σ) ∈ T n (q), the set is a basis for Z S,σ /D S,σ .
Proof. Proceed in two steps, based on the division of Z S,σ /D S,σ into the subquotients ut ↑ 1 (λ) /D S,σ and Z S,σ /ut ↑ 1 (λ) . First, if S = λ ν, then Describing a basis for Z S,σ /ut ↑ 1 (λ) is more straightforward. The subspace is a transversal of Z S,σ /ut ↑ 1 (λ) , and the set {b C | C ∈ cols(S)} is a basis for this subspace.

Labeled loopless binary matroids and ideals
In this section, fix λ ∈ NNSP n and (S, σ) ∈ T λ (q). Recall that CR(S) = cols(S) rows(S). Section 3.4 describes a total order on the set CR(S). With this order, the construction in Section 2.3 describes the set → G CR(S) (q) of F × q -labeled loopless binary matroids on CR(S). Each (M, τ ) ∈ → G CR(S) (q) will determine an ideal of ut n . If M = (U, V, E), let The subspace n S,σ,M,τ satisfies D S,σ ⊆ n S,σ,M,τ ⊆ Z S,σ , so by Lemma 4.8 is an ideal of ut n . Example 4.14. For n = 6, and q = 5, take S = λ ν, σ : bind(S) → F × 5 , cols(S) = {C 1 , C 2 }, and rows(S) = {R 1 } as given by The canonical order on CR(S) is C 1 < C 2 < R 1 , and the set of loopless binary matroids on an isomorphic ordered set is given in Example 2.4. Below are three elements of → G CR(S) (5) with the corresponding members of fam(S, σ): and In order to prove Proposition 4.13, we will view Z S,σ /D S,σ as the set of CR(S)-indexed coordinates from F q , as in (α C 1 , . . . , α C | cols(S)| , α R 1 , . . . , α R | rows(S)| ) = V∈CR(S) Given any subspace W of Z S,σ /D S,σ , there is a canonical basis for W which will serve as a stand-in for W itself. Let k = dim(W ) and take (x i,V ) 1≤i≤k,V∈CR(S) to be the unique reduced row-echelon form matrix over F q with row span W . That is, if then W = F q -span{x 1 , . . . , x k } and there are elements U 1 < U 2 < · · · < U k of CR(S) with Call this basis the RRE basis of W . Proof. Let n be the unique subspace of Z S,σ with D S,σ ⊆ n and n/D S,σ = W . By Lemma 4.8, n is an ideal of ut n . Recall that n ∈ fam(S, σ) if and only if for each (i, j) ∈ ↑ 1 (λ), e i,j ∈ n if and only if (i, j) / ∈ ν, and (iii) n ⊆ Z S,σ .
By assumption, (iii) holds. The following shows that (i) is equivalent to (a), and (ii) to (b).
For each (r, s) ∈ λ, the (r, s)-entry of each member of any coset in ut λ /ut ↑ 1 (λ) will be identical. As D S,σ ⊆ ut ↑ 1 (λ) , the same is true for ut λ /D S,σ . Therefore, for each a ∈ Z S,σ , C ∈ cols(S), and (r, s) ∈ C, the (r, s)-entry of each member of a + D S,σ is nonzero if and only if the C-coordinate of a + D S,σ is nonzero. Since cols(S) partitions the set λ, (i) is equivalent to (a).
For R ∈ rows(S) and (i, j) ∈ R,  To show that the given map is also a bijection, construct an inverse as follows. For n ∈ fam(S, σ), let W = n/D S,σ and take {x i | 1 ≤ i ≤ k} to be the RRE basis of W with corresponding indices U 1 < U 2 < · · · < U k satisfying conditions 1. and 2. above. Let

Normal subgroups
Recall that Theorem A asserts a correspondence between the normal subgroups of UT n (F q ) and certain ideal-like additive subgroups of ut n (F q ), and that as a consequence the results of Section 4 also apply to certain normal subgroups, giving Theorem B. One particularly nice consequence is the following corollary. A number of examples which apply Theorem 4.1 to construct previously obscure ideals (and thus normal subgroups) can be found in Section 4. The indexing for the better known normal subgroups of UT n (F q ) in Corollary 5.1 is similar but distinct from previous work.
In [22,Theorem 4.1], Marberg constructs a family of normal subgroups of UT n (F p ) which have the form 1 + a for a two-sided associative algebra ideal a ⊆ ut n (F p ). These ideals are nicely characterized as the subspaces a ⊆ ut n (F p ) with More generally, these subgroups are indexed by tuples (λ, f ∅ , M, τ ), where λ = λ ∅ denotes the trivial splice of λ ∈ NNSP n , f ∅ is the unique labeling of bind(λ) = ∅, and (M, τ ) is an F × p -labeled loopless binary matroid on the set CR(λ), which contains only one-element columns.
Some caution is required when comparing Corollary 5.1 to [22,Theorem 4.1], however, as the latter uses a different convention for relating nonnesting set partitions to subsets of ut n (F q ). In this paper, a nonnesting set partition determines the support of each corresponding subgroup, but in [22] nonnesting set partitions determine a different property which is, in a sense, dual to support. The dual property is difficult to describe in the scope of this paper, but the indexing schemes are morally equivalent (see [2,Section 3.1] for details), and versions of both results can be given in either convention.
The remainder of the section is divided into two subsections. Subsection 5.1 gives a proof of Theorem A, which is not significantly different from that of [20,Theorem 1]. This is included mainly for completeness, but some intermediate results are needed in the sequel. Subsection 5.2 states new results about the normal subgroups of UT n (F q ), including an enumerative formula and a discussion of the lattice of normal subgroups.

Proof of Theorem A
In this section fix a prime power q, and write ut n = ut n (F q ) and UT n = UT n (F q ). In order to distinguish the Lie bracket from the group commutator, I will adopt the notation The following computation will be used repeatedly: for a ∈ UT n , (s, t) ∈ [[n]], and x ∈ F q , [1 + xe s,t , a] = 1 + x (r,u) (s,t) (a −1 ) r,s a t,u e r,u . Proof. By assumption N ⊆ UT λ , so [UT n , [UT n , N ]] ⊆ UT ↑ 2 (λ) . The opposite containment will follow from the fact that which generates UT ↑ 2 (λ) . Fix (i, l) ∈ ↑ 2 (λ) and x ∈ F q . It is possible to choose • (j, k) ∈ λ with (j, k) (i, l) and ht((j, k)) + 2 ≤ ht((i, l)) and • a ∈ N with a j,k = 0. The (i, l)-entry of b is xa j,k = 0, so a standard computation (e.g. [10,Corollary 3.4]) shows that b is conjugate to 1 + xa j,k e i,l .
Elements of the form 1 + xe j,j+1 generate UT n , and from consideration of support is an isomorphism, completing the proof.
Lemma 5.4. Let λ ∈ NNSP n and S ∈ T λ . The bijection is an isomorphism of groups.
Proof. By consideration of support the statement will follow if Suppose that (i, j), (j, k) ∈ ↑ (λ), and note that ht((i, k)) = ht((i, j)) + ht((j, k)). If either ht((i, j)) > 1 or ht((j, k)) > 1, then (i, k) ∈ ↑ 2 (λ), a subset of ↑ (λ) − S. Otherwise (i, j), (j, k) ∈ λ, in which case i and k are in the same connected component of the graph of λ. In this case (i, k) is not contained in any splice of λ by Lemma 3.1, so (i, k) ∈ ↑ (λ) − S. Lemma 5.5. Let n be a subgroup of ut n with {[a, b] Lie | a ∈ ut n , b ∈ n} ⊆ n, and let n = F q -span(n). Then 1. [ut n , n] Lie = [a, b] Lie | a ∈ ut n , b ∈ n , and this is a subset of n, and 2. n is an ideal of ut n .
Proof. Take a ∈ ut n , b 1 , b 2 , . . . , b k ∈ n, and α 1 , α 2 , . . . , α k ∈ F q . Then a, Proof of Theorem A. Let N ⊆ UT n and n ⊆ ut n with N = 1 + n, and take λ ∈ NNSP n with ↑ (λ) = supp(N ) = supp(n). The aim is to show that N is a normal subgroup if and only if n is an additive subgroup with {[a, b] Lie | a ∈ ut n , b ∈ n} ⊆ n.
First suppose that N UT n . Let m = [UT n , N ] − 1. By Lemma 5.2 and Lemma 5.3, and m ⊆ n, so what remains is to show that n is an additive group. By equation (5.2), m and n meet the conditions of Lemma 4.6, so there is a tight splice S of λ for which Under the isomorphism of Lemma 5.4, n/ut ↑(λ)−S maps to N/UT ↑(λ)−S , so n is a group. Now suppose that n is a subgroup of ut n with {[a, b] Lie | a ∈ ut n , b ∈ n} ⊆ n. By Lemma 5.5, n = F q -span(n) is an ideal and [ut n , n] Lie ⊆ n. By Lemma 4.6, there is a tight splice S of λ for which ut ↑(λ)−S ⊆ [ut n , n] Lie ⊆ n.
Under the isomorphism of Lemma 5.4, N/UT ↑(λ)−S is the image of n/ut ↑(λ)−S , and so N is a subgroup of UT n . Lemma 5.3 then gives [UT n , N ] ⊆ [a, b] | a ∈ ut n , b ∈ n + ut ↑ 2 (λ) ⊆ N, and so N UT n .
A slightly stronger restatement of Lemma 5.3 now follows.
Proof. This follows from Lemma 5.2, Lemma 5.3, and Theorem A.

Further results on normal subgroups
Some aspects of my description of Lie algebra ideals extend to the set {N UT n (F q )}, even when q is not prime. For (S, σ) ∈ T n (q) with S = λ ν, define the normal subgroup family NSGfam(S, σ).
Lemma 4.11 states that for any (S, σ) ∈ T n (q), Z S,σ /D S,σ = F q -span{b V + D S,σ | V ∈ CR(S)}. Proof. Take n ≤ ut n (F q ) so that D S,σ ⊆ n and n/D S,σ = W . As n ⊆ Z S,σ , what remains is to determine when F q e i,j ⊆ n for each (i, j) ∈ ν and supp(n) = ↑ (λ), where S = λ ν. These conditions are respectively equivalent to W satisfying (Int) for all R ∈ rows(S) and all C ∈ cols(S).
(R3) Computing |{N UT n (F q )}| for 1 ≤ n ≤ 10 and 1 ≤ d ≤ 5 suggests that this quantity may have positive and unimodal coefficients as a polynomial in p − 1. Corollary 4.2 implies positivity for d = 1 and n ≥ 0, but unimodality is entirely mysterious.

Lattices of normal subgroups
The ideals of ut n (F q ) naturally form a lattice, so it is also possible to endow the set I n (q) = {1 + n | n is an ideal of ut n (F q )} with a lattice structure: (1 + n) ∧ (1 + m) = 1 + (n ∩ m) and (1 + n) ∨ (1 + m) = 1 + (n + m).
The set {N UT n (F q )} is also a lattice, with ∧ and ∨ given respectively by intersection and product of subgroups. A consequence of Theorem A is that these lattice structures agree.