Polytopal models and tropical geometry

Arkady Berenstein

Plan:

• Tensor product multiplicities

• Crystal bases Let G^{∨} be a complex reductive group such as GL_{3}(ℂ). Then we have

Δ_{λ} = Δ_{1×1}^{λ_{1}−λ_{2}} · Δ_{2×2}^{λ_{2}−λ_{3}} · Δ_{3×3}^{λ_{3}} ∈ ℂ[G^{∨}]

for the highest weight vector and V_{λ} = Span{Δ_{λ} ◁ G^{∨}}. Problem. Compute the tensor product multiplicity c_{λμν} = dim(V_{λ} ⊗ V_{μ} ⊗ V_{ν})^{G^{∨}}. Theorem (A. Zelevinsky, A. Berenstein, ∼1999). There is a convex polyhedral cone C\underline{i} ⊂ V = ℝ^{|R_{+}^{∨}|+2ℓ}, where \underline{i} is a reduced decomposition of the longest element in the Weyl group (for GL_{3}(ℂ), \underline{i} = (1 2 1) or \underline{i} = (2 1 2)), R_{+}^{∨} is the set of positive roots, and ℓ = rank[G^{∨} : G^{∨}], and a linear map π\underline{i} : V → (P^{∨} ⊗ ℝ)^{3} such that

|π−1 ; \underline{i}(λ, μ, ν) ∩ (c\underline{i})_{ℤ}| = c_{λμν}.

The associated picture is

•  (λ, μ, ν) C ⊂ ℝ^{3ℓ}  fiber, of dimension |ℝ_{+}^{∨}| − ℓ

Now let G^{∨} = GL_{3}, λ = (λ_{1}, λ_{2}, λ_{3}) be non-decreasing, and c_{λμ}^{ν} = mult(V_{μ}, V_{λ} ⊗ V_{ν}) = c_{λμ^{∗}ν}.

Theorem (I. M. Gelfand, A. Zelevinsky, ∼1985). The number c_{λμ}^{ν} is the number of all triples (t̃_{1}, t̃_{2}, t̃_{3}) ∈ (ℤ_{≥0})^{3} such that

λ_{1} − λ_{2} ≥ t̃_{1},  λ_{2} − λ_{3} ≥ t̃_{2}  λ_{2} − λ_{3} ≥ t̃_{2} + t̃_{3} − t̃_{1},
ν_{1} − ν_{2} ≥ t̃_{2},  ν_{1} − ν_{2} ≥ t̃_{2} + t̃_{1} − t̃_{3},  ν_{2} − ν_{3} ≥ t̃_{3},
λ + ν − μ = t̃_{1}α_{1}^{∨} + t̃_{2}(α_{1}^{∨} + α_{2}^{∨}) + t̃_{3}α_{2}^{∨} = (t̃_{1} + t̃_{2})α_{1}^{∨} + (t̃_{2} + t̃_{3})α_{3}^{∨},  (1)

whre the α_{i}^{∨} are the co-roots.

Remarks.

1. For ν_{1} − ν_{2}, ν_{2} − ν_{3} large, we obtain the Kostant partition function. In fact,

c_{λν}^{μ} ≤ K_{λ,μ−ν} ≤ ρ(λ + ν − μ).

2. In this case, we can produce a GT-pattern:

λ_{1}  λ_{2}  λ_{3} λ_{2} + t̃_{1}  λ_{3} + t̃_{2} λ_{3} + t̃_{2} + t̃_{3}

Note that by the inequalities given in the theorem, we can “move down” the diagonals from left to right via ≥: λ_{1} ≥ λ_{2} + t̃_{1} ≥ λ_{3} + t̃_{2} + t̃_{3} and λ_{2} ≥ λ_{3} + t̃_{2}. Similarly, since (t̃_{1}, t̃_{2}, t̃_{3}) ∈ (ℤ_{≥0})^{3}, we can “move down” the diagonals from right to left via ≤: λ_{2} ≤ λ_{2} + t̃_{1} and λ_{3} ≤ λ_{3} + t̃_{2} ≤ λ_{3} + t̃_{2} + t̃_{3}. (For B_{n}, we can take the longest word for B_{n−1}, and extend it uniquely to B_{n} to form analogous GT-patterns.)

3. If we set ν_{1} − ν_{2} = ∞ and ν_{2} = ν_{3}, then we obtain the multiplicities for the Levi subgroup. This works in every Levi subgroup.

We now use tropical geometry to geometrize and generalize. Rewrite ν − μ in (1):

λ_{2} − λ_{3} + min {−t̃_{2}, t̃_{1} − t̃_{2} − t̃_{3}}

Then in the (min, +)-algebra and writing [t_{i}] = t̃_{i}, [a+b] = min {ã, b̃}, and [ab] = ã+b̃ (which

implies [\frac{1}{a}] = −ã), we can rewrite the inequalities from the theorem as follows (“outside the brackets is PL combinatorics; inside is ordinary algebraic geometry with no minus”):

λ_{1} − λ_{2} + [\frac{1}{t_{1}} ] ≥ 0,  λ_{2} − λ_{3} + [\frac{t_{1} + t_{3}}{t_{2}t_{3}} ] ≥ 0,
ν_{1} − ν_{2} + [\frac{t_{1} + t_{3}}{t_{1}t_{2}} ] ≥ 0,  ν_{2} − ν_{3} + [\frac{1}{t_{3}} ] ≥ 0
λ + ν − μ = [t_{1}t_{2}]α_{1}^{∨} + [t_{2}t_{3}]α_{2}^{∨}.

We can now interpret the expressions in brackets as functions on the upper triangular part of G. For now,

G = GL_{3}, U = \{(\begin{matrix}1 & ∗ & ∗ \\ 0 & 1 & ∗ \\ 0 & 0 & 1 \end{matrix})\} .

Let the coordinate chart on the unipotent group U be given by

x_{1}(t) = (\begin{matrix}1 & t & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) , x_{2}(t) = (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & t \\ 0 & 0 & 1 \end{matrix}) .

Then

U = x_{1}(t_{1})x_{2}(t_{2})x_{3}(t_{3}) = (\begin{matrix}1 & t_{1} + t_{3} & t_{1}t_{2} \\ 0 & 1 & t_{2} \\ 0 & 0 & 1 \end{matrix}) .

Write

Δ_{1} = t_{1}t_{2}, Δ_{2} = det (\begin{matrix}t_{1} + t_{3} & t_{1}t_{2} \\ 1 & t_{2} \end{matrix}) = t_{2}t_{3}.

In general,

Δ_{i} = det  i  i  Δ′_{i} = det  i − 1 i  11′Δ_{i} = det _{1 1 i − 1 i}

Thus,

Δ′_{1} = t_{2},  Δ′_{2} = det (\begin{matrix}t_{1} + t_{3} & t_{1}t_{2} \\ 0 & 1 \end{matrix}) = t_{1} + t_{3},
\frac{Δ′_{1}}{Δ_{1}} = [\frac{1}{t_{1}} ] ,  \frac{Δ′_{2}}{Δ_{2}} = [\frac{t_{1} + t_{3}}{t_{2}t_{3}} ] ,
\frac{′Δ_{1}}{Δ_{1}} = [\frac{t_{1} + t_{3}}{t_{1}t_{2}} ] ,  \frac{′Δ_{2}}{Δ_{2}} = [\frac{1}{t_{3}} ] ,
Δ_{1} = [t_{1}t_{2}]  Δ_{2} = [t_{2}t_{3}].

Generalizing, let G be an algebraic reductive group, B a Borel subgroup of G, and U the unipotent subgroup of G. Let {i} be the vertices of the Dynkin diagram. Put 𝔾_{m} = ℂ^{×}, and consider

x\underline{i} : (𝔾_{m})^{N} → U_{0}
(t_{1}, …, t_{N}) ↦ x_{i_{1}}(t_{1}) ⋯ x_{i_{N}} (t_{N}),

where N = dim U.

Theorem (A. Zelevinsky, Fomin, A. Berenstein). If \underline{i} is a reduced word for w_{0}, then x\underline{i} is an open inclusion (𝔾_{m})^{N} ↪ U_{0}, where U_{0} = U ∩ B_{−}w_{0}B_{−}.

Aside: Cluster algebras are such charts that cover a target variety.

Define a principle minor Δ_{λ} ∈ ℂ[G] by

Δ_{λ}(g) = λ(h),

where g = U_{−}hU_{+}, the character of the diagonal part.

Example. Let G = GL_{n}. Then

λ = (\underbrace{1, …, 1}_{i}, 0, …, 0),  and  Δω_{i} = det  ^{i  i}

Note that Δ_{ω_{i}} is the ordinary principal minor. For w, σ ∈ W and ω_{i} a fundamental weight,

Δ_{wω_{i},σω_{i}}(g) = Δ_{ω_{i}} (\overline{w}^{−1}g\overline{σ}) .

If w = s_{i_{1}} ⋯ s_{i_{m}}, then \overline{w} = \overline{s_{i_{1}}} ⋯ \overline{s_{i_{m}}}, where

\overline{s_{i}} = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix})_{i} .

In this larger context,

Δ_{i} = Δ_{ω_{i},ω_{0}ω_{i}}, Δ′_{i} = Δ_{s_{i}ω_{i},ω_{0}ω_{i}},
′Δ_{i} = Δ_{ω_{i},ω_{0}s_{i}ω_{i}},

and

C\underline{i} = \{t̃∈ (ℤ_{≥0})^{dim U} : (λ, α_{i}) + [\frac{Δ′_{i}}{Δ_{i}} ] ≥ 0, (ν, α_{i^{∗}}) + [ \frac{′Δ_{i}}{Δ_{i}} ] ≥ 0,
λ + ν − μ = Σ [Δ_{i}]α_{i}^{∨}} ,

where α_{i^{∗}} = −ω_{0}(α_{i}).

Returning to the Levi discussion, we have

c_{λμ}^{ν} = dim V_{λ}(μ − ν, ν),

where V_{λ}(μ − ν, ν) = {v ∈ V_{λ}(μ − ν) : e^{(ν,α_{i}^{∨})+1}(v) = 0}. Choose a subset I of the Dynkin diagram. We want ν such that

(ν, α_{i}^{∨}) = \begin{cases}0 & if i ∉ I, \\ ∞ & if i ∈ I, \end{cases}

where ∞ stands for a very large number. Looking at the Levi L = ⟨x_{i}(… ⟩, we have

mult(V_{β} : V_{λ}|_{L}) = dim Hom_{L}(V_{β}, V_{λ}).

This works even for Kac-Moody.

Key papers about this:

• tensor product, piecewise (Inventiones, 1999)

• Kazdan poasted years ago to arXiv geometric crystals II