GL-multiplicities of A_n-quiver loci

What this computes

Fix an A_n quiver Q with any orientation, a dimension vector d, and a quiver locus Ω ⊆ rep_Q(d) (an orbit closure, given by its rank conditions). The coordinate ring C[Ω] decomposes under GL(d) = ∏ GL_di as ⊕_Λ V_Λ^⊕m. The theorem of Cavey–Hardt–Yong computes m_Ω,Λ as the number of highest-weight Ω-admissible matrices M of profile Λ. This applet enumerates exactly those.

Sharing this page

The file is completely self-contained — no server, no libraries, no network. To send it to someone, just send them the file; they save it and double-click it. To send a link instead, drop the single file on any web space (a personal page, or a GitHub repo with Pages enabled) and send that URL. copy link in the top right copies the current address including a code for the exact quiver, orbit and profile on screen, so a colleague opening it lands on the same example.

The Q-shape

Replacing Q by a bipartite quiver (inserting an identity map at every vertex whose two arrows point the same way) arranges the matrices M₁,…,M_n−1 into a staircase of regions R₁,…,R_m running from southwest to northeast. The shaded, dashed blocks are the phantom regions: they carry no entries of M, but every one of their cells counts as occupied when antidiagonals are measured.

Admissible

M is Ω-admissible when, for every 1 ≤ i ≤ j ≤ m, the part of M lying in R_i∪⋯∪R_j contains no antidiagonal of length r_[i,j]+1. An antidiagonal is a sequence of positions with strictly increasing rows and strictly decreasing columns, each of which either holds a positive entry or lies in a phantom region. The audit panel under each matrix shows the longest antidiagonal against r_[i,j] for every region range.

Highest weight and the profile

At each vertex j read the word read_j(M_j−1)·read_j(M_j), taking the row word row(M) when j is the tail of the edge, and the reverse complement word col*(M) when j is the head. M is highest weight when RSK-inserting all n words gives supersemistandard tableaux (only j's in row j). Their shapes λ^(j), together with Δ_j = the total of the matrices having j as head, form the profile.

Panel 2 · the orbit

Ω is an orbit closure, so it is determined by any one of three equivalent things, all shown: the partial permutation matrices L_e of a representative (panel 1); the rank array r_[i,j], the rank of the filling of R_i∪⋯∪R_j; and the orbit type, the decomposition of the representative into Gabriel's indecomposables V_[a,b] (one-dimensional at each vertex from a to b, identity maps between them). Editing a rank-array entry updates the orbit type and the representative to match. Not every array of numbers is the rank array of an orbit; when the one on screen is not, the applet says so and dims what is below it (which still describes the last valid orbit) but keeps what you typed, so you can go on to the other entries you meant to change. revert puts the last valid array back. Panel 3 will not compute until the array is an orbit again.

variety of complexes (equioriented quivers) is driven by the ranks r_i of the individual maps instead: it starts from an example with genuine rank conditions, and the orbit is rebuilt as you type — there is nothing to press. random ranks picks another admissible set.

Panel 3 · what to compute

• one profile Λ — type Λ (in any of the three encodings below) and press Enumerate all to get m_Ω,Λ and every matrix counted by it. random Λ fills the fields with a profile that is guaranteed to occur; it does not run the search, so the flow is the same whether you typed Λ or not. It chooses a degree for you — roughly |M_e| ≈ min(d_tail, d_head), which is calibrated to be interesting without making the follow-up enumeration slow. Type a degree on the a whole degree tab first and it will use that instead.
• One random matrix stops at the first matrix it finds instead of enumerating. This is the tool for cases too large to enumerate — it is milliseconds even when the full count is astronomical. The sample is random but not uniform.
• advanced ▸ highest-weight matrices only, when switched off, browses all of Adm(Ω). Over a whole degree these are grouped into crystal components: each block is one irreducible V_Λ, the starred matrix is its highest-weight element, everything else in the block is reached from it by lowering operators, and the number of blocks is the multiplicity. (With a profile fixed instead, the filter is on the weight, so a component would be cut down to its highest vector; the applet says so and does not group.)
• a whole degree takes a multidegree (one number per edge) or a single total degree and lists every irreducible occurring there with its multiplicity. Click a row to enumerate it.

Entering a profile — three equivalent ways

• GL-weight — the honest dominant weight, d weakly decreasing integers.
• [λ, Δ] — the straight shape and the determinant twist, so V = V_λ(d) ⊗ det^−Δ. Leave Δ blank and it is deduced from the shapes when possible.
• μ ÷ ν — Stembridge's rational shape (write 2,1 ÷ 1), which needs ℓ(μ)+ℓ(ν) ≤ d.

Switching between the three converts what you have already typed. The multidegree (|M_e|) and the twists Δ are always determined by Λ, and are shown under the editor.

Reading the output

Results open as a grid of thumbnails, headed by Λ drawn as rational shapes; click a thumbnail for the full view, and grid to come back. ← → step through them. The detail panel lists the matrices M_e, the reading word and RSK tableau at every vertex (with the separated words read_Q(M) underneath), the profile in all three encodings together with the T(d)-weight wt(M) that the character formula sums, and the rank audit.

Two further panels open on demand. rational tableaux shows quiverRSK(M) = ins(read_Q(M)), the pairs U ÷ V produced by Stembridge/Stroomer insertion; their rational shapes are the profile, and M is highest weight exactly when every U is supersemistandard and every V lowest weight. crystal gives the local structure at this matrix: the dimension of its component (= dim V_Λ by the hook-content formula, so the number of admissible matrices sharing its highest weight), the highest-weight matrix itself, and every neighbour E_i^(j), F_i^(j) with the string lengths ε and φ. Click any of those little diagrams to walk to that matrix; back retraces your steps and back to result #k returns you to where you were in the list. Walking never scrolls the page: the panel is redrawn where it is.

Everything that folds — these panels and the two ▸ sections in panels 2 and 3 — starts closed each time the page is opened; what you open then stays open as you step through the matrices.

When a search takes a while

A progress bar appears with the elapsed time and the number of search nodes, and Stop ends it immediately. The time limit (default 15 s) is under advanced, together with how many matrices to keep and the highest-weight switch. If a search is cut short the applet says so and the count is reported as a lower bound.

Special cases worth trying

• Variety of complexes: equioriented, then edit the ranks r_i (they start at an example that is not the whole space, and the orbit follows as you type) — every multiplicity is 1 and the profiles are exactly (λ₁÷∅, λ₂÷λ₁, …, ∅÷λ_n−1).
• Bipartite A₃ C^p→C^q←C^r with no rank restrictions: the multiplicities are Littlewood–Richardson coefficients.
• No rank restrictions: C[rep_Q(d)] is multiplicity free exactly when d avoids (a,b,c) with a≥2, b≥3, c≥2 and (a,b,c,d) with a≥1, b≥2, c≥2, d≥1.