An introduction to crystals


Informally, a crystal \(\mathcal{B}\) is an oriented graph with edges colored in some set \(I\) such that, for each \(i\in I\), each node \(x\) has:

  • at most one \(i\)-successor, denoted \(f_i x\);
  • at most one \(i\)-predecessor, denoted \(e_i x\).

By convention, one writes \(f_i x=\emptyset\) and \(e_i x=\emptyset\) when \(x\) has no successor resp. predecessor.

One may think of \(\mathcal{B}\) as essentially a deterministic automaton whose dual is also deterministic; in this context, the \(f_i\)‘s and \(e_i\)‘s are respectively the transition functions of the automaton and of its dual, and \(\emptyset\) is the sink.

A crystal comes further endowed with a weight function \(\operatorname{wt}: \mathcal{B}\mapsto L\) which shall satisfies appropriate conditions.

In combinatorial representation theory, Crystals are used as combinatorial data to model representations of Lie algebra.

Axiomatic definition

Let \(C\) be a CartanType with index set \(I\), and \(L\) be a realization of the weight lattice of the type \(C\). Let \(\alpha_i\) and \(\alpha^{\vee}_i\) denote the simple roots and coroots respectively.

A type \(C\) crystal is a non-empty set \(\mathcal{B}\) endowed with maps \(\operatorname{wt} : \mathcal{B} \to L\), \(e_i, f_i : \mathcal{B} \to \mathcal{B} \cup \{\emptyset\}\), and \(\varepsilon_i, \varphi_i : \mathcal{B} \to \ZZ \cup \{-\infty\}\) for \(i \in I\) satisfying the following properties for all \(i \in I\):

  • for \(b, b^{\prime} \in \mathcal{B}\), \(f_i b^{\prime} = b\) if and only if \(e_i b = b^{\prime}\);
  • if \(e_i b \in \mathcal{B}\), then:
    • \(\operatorname{wt}(e_i x) = \operatorname{wt}(b) + \alpha_i\),
    • \(\varepsilon_i(e_i b) = \varepsilon_i(b) - 1\),
    • \(\varphi_i(e_i b) = \varphi_i(b) + 1\),
  • if \(f_i b \in \mathcal{B}\), then:
    • \(\operatorname{wt}(f_i b) = \operatorname{wt}(b) - \alpha_i\),
    • \(\varepsilon_i(f_i b) = \varepsilon_i(b) + 1\),
    • \(\varphi_i(f_i b) = \varphi_i(b) - 1\),
  • \(\varphi_i(b) = \varepsilon_i(b) + \langle \alpha^{\vee}_i, \operatorname{wt}(b) \rangle\),
  • if \(\varphi_i(b) = -\infty\) for \(b \in \mathcal{B}\), then \(e_i b = f_i b = \emptyset\).

Some further conditions are required to guarantee that this data indeed models a representation of a Lie algebra. For finite simply laced types a complete characterization is given by Stembridge’s local axioms [St2003].

REFERENCES:

[St2003]J. Stembridge, A local characterization of simply-laced crystals, Trans. Amer. Math. Soc. 355 (2003), no. 12, 4807-4823.

EXAMPLES:

We construct the type \(A_5\) crystal on letters (or in representation theoretic terms, the highest weight crystal of type \(A_5\) corresponding to the highest weight \(\Lambda_1\)):

sage: C = crystals.Letters(['A',5]); C
The crystal of letters for type ['A', 5]

It has a single highest weight element:

sage: C.highest_weight_vectors()
(1,)

A crystal is an enumerated set (see EnumeratedSets); and we can count and list its elements in the usual way:

sage: C.cardinality()
6
sage: C.list()
[1, 2, 3, 4, 5, 6]

as well as use it in for loops:

sage: [x for x in C]
[1, 2, 3, 4, 5, 6]

Here are some more elaborate crystals (see their respective documentations):

sage: Tens = crystals.TensorProduct(C, C)
sage: Spin = crystals.Spins(['B', 3])
sage: Tab  = crystals.Tableaux(['A', 3], shape = [2,1,1])
sage: Fast = crystals.FastRankTwo(['B', 2], shape = [3/2, 1/2])
sage: KR = crystals.KirillovReshetikhin(['A',2,1],1,1)

One can get (currently) crude plotting via:

sage: Tab.plot()
Graphics object consisting of 52 graphics primitives

If dot2tex is installed, one can obtain nice latex pictures via:

sage: K = crystals.KirillovReshetikhin(['A',3,1], 1,1)
sage: view(K, pdflatex=True, tightpage=True) #optional - dot2tex graphviz

or with colored edges:

sage: K = crystals.KirillovReshetikhin(['A',3,1], 1,1)
sage: G = K.digraph()
sage: G.set_latex_options(color_by_label = {0:"black", 1:"red", 2:"blue", 3:"green"}) #optional - dot2tex graphviz
sage: view(G, pdflatex=True, tightpage=True) #optional - dot2tex graphviz

For rank two crystals, there is an alternative method of getting metapost pictures. For more information see C.metapost?.

Todo

  • Vocabulary and conventions:
    • For a classical crystal: connected / highest weight / irreducible
    • ...
  • Layout instructions for plot() for rank 2 types
  • RestrictionOfCrystal

The crystals library in Sage grew up from an initial implementation in MuPAD-Combinat (see <MuPAD-Combinat>/lib/COMBINAT/crystals.mu).

class sage.combinat.crystals.crystals.CrystalBacktracker(crystal, index_set=None)

Bases: sage.combinat.backtrack.GenericBacktracker

Time complexity: \(O(nF)\) amortized for each produced element, where \(n\) is the size of the index set, and \(F\) is the cost of computing \(e\) and \(f\) operators.

Memory complexity: \(O(D)\) where \(D\) is the depth of the crystal.

Principle of the algorithm:

Let \(C\) be a classical crystal. It’s an acyclic graph where all connected component has a unique element without predecessors (the highest weight element for this component). Let’s assume for simplicity that \(C\) is irreducible (i.e. connected) with highest weight element \(u\).

One can define a natural spanning tree of \(C\) by taking \(u\) as the root of the tree, and for any other element \(y\) taking as ancestor the element \(x\) such that there is an \(i\)-arrow from \(x\) to \(y\) with \(i\) minimal. Then, a path from \(u\) to \(y\) describes the lexicographically smallest sequence \(i_1,\dots,i_k\) such that \((f_{i_k} \circ f_{i_1})(u)=y\).

Morally, the iterator implemented below just does a depth first search walk through this spanning tree. In practice, this can be achieved recursively as follow: take an element \(x\), and consider in turn each successor \(y = f_i(x)\), ignoring those such that \(y = f_j(x^{\prime})\) for some \(x^{\prime}\) and \(j<i\) (this can be tested by computing \(e_j(y)\) for \(j<i\)).

EXAMPLES:

sage: from sage.combinat.crystals.crystals import CrystalBacktracker
sage: C = crystals.Tableaux(['B',3],shape=[3,2,1])
sage: CB = CrystalBacktracker(C)
sage: len(list(CB))
1617
sage: CB = CrystalBacktracker(C, [1,2])
sage: len(list(CB))
8

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