Simple Proof of Tokuyama’s Formula
Tokuyama’s Formula is a combinatorial result that expresses the product of the deformed Weyl denominator and the Schur polynomial as a sum over strict Gelfand-Tsetlin patterns. This result implies Gelfand’s parametrization of the Schur polynomial, Weyl’s Character Formula, and Stanley’s formula on Hall-Littlewood polynomials — all for ; also, the formula is related to alternating sign matrices. Moreover, Tokuyama’s formula gives a combinatorial evaluation of certain matrix coefficients of representations of
-adic groups.
A Gelfand-Tsetlin pattern (or GT-pattern for short) with the top row is a triangular array of numbers
such that the betweenness condition is satisfied for all
. In other words, each entry in the pattern lies between two entries above it. A Gelfand-Tsetlin pattern is called strict if rows are strictly decreasing, that is,
for all
. Denote by
and
the sets of GT patterns and strict GT patterns with the top row
, respectively.
We call entry left-leaning if
. In other words, the entry equals to the entry above on the left. Similarly,
is right-leaning if
. We call
generic if it’s not left- or right-leaning.
Let and
be the number of left- and right-leaning entries in GT-pattern
, respectively. Let
be the number of generic entries. Also, let
be the
-th row sum, and
, the difference of sums of consequent rows. Now we can state
Theorem. (Tokuyama, 1988). Let be a partition of length
. We have
where , and
is the Schur polynomial.
The formula expresses a weighted sum over strict Gelfand-Tsetlin patterns in the closed form as the product of the deformed Weyl denominator and the Schur polynomial. Note that the usual Weyl denominator for
is obtained by setting
.
Example. Let’s see Tokuyama’s formula in action. Let . Then
. In the picture above you can see all the patterns
with the corresponding contribution to the sum. Yellow entries are left-leaning, blue entries are generic.
Now, if we sum all the contributions, we get
Beautiful. Before going to the proof of Tokuyama’s formula, let’s have a look at its multiple corollaries.
Corollary. (Gelfand’s parametrization) Let be a partition of length
. Then
Sketch of the proof. Set , then non-zero terms in the sum are exactly strict Gelfand-Tsetlin patterns with the top row
with no right-leaning terms. They are in bijection with all Gelfand-Tsetlin patterns with the top row
. QED.
Corollary. (Weyl’s Character Formula). Let be a partition of length
. Let
be functions from the bialternant formula of Jacobi. Then
Sketch of the proof. Set , then non-zero terms in the sum are exactly strict Gelfand-Tsetlin patterns with the top row
with no generic terms. They are in bijection with the permutation group
. The sum becomes the deformed Vandermonde determinant since
equals to the number of inversions in
. QED.
In a similar fashion, by setting , one can prove Stanley’s formula for expressing Hall-Littlewood polynomial as a sum over strict Gelfand-Tsetlin patterns. See the details in Section 3.3. Another connection is with alternating sign matrices: the sum in Tokuyama’s formula can be interpreted as a sum over alternating sign matrices which gives a generalization of the Weyl’s Character Formula where the sum is taken just over permutations (which are a subset of the alternating sign matrices). See Section 3.4. for details.
The original proof in terms of characters of tensor products of highest weight representations by Tokuyama is a little hard to read. A better proof can be given in terms of solvable lattice models, but I will show the following short proof by Daniel Bump which is expressed in a more familiar language.
Denote . Note that in a strict GT pattern each row is strictly decreasing, and so it can be written as
for some partition
of length
. Recall that partition
of length
interleaves partition
of length
if
Note that each row in a GT pattern interleaves the next row. In a strict GT pattern every row interleaves
above it.
Idea of the proof. Denote the left and right sides of Tokuyama’s Formula by
We will show the following branching rules for and
:
where partitions are such that
interleaves with
, and
interleaves with
; also,
is a vertical strip. Then we show that
, and by induction we are done.
Proof. We start with . We consider the contribution of patterns
with top row
and second row
. Since
interleaves
we have
, or
. Removing the top row from
produces a pattern with the top row
and collecting these produces
with a coefficient equal to
. Thus, we proved the branching rules for
.
Next, consider . Recall the classical branching rules for the Schur polynomial:
where partitions of length
interleaves with
. We also have
where is the
-th elementary symmetric polynomial. Then, replace
and
in
to get that
equals
Recall Pieri’s formula in the form , where each
is such that
is a
-vertical strip, that is,
or
, and
exactly
times. Collecting everything together, we get that
equals to
with as above. Next, we notice that
and
, and that
interleaves
since
interleaves
and
is a vertical strip. Hence, we can write the branching rules for
as follows:
Finally, we have to prove that
where interleaves
and
is a vertical strip. The conditions on
can be written as
and
or
. Hence, we can write the right side in terms of individual
as follows:
Now, if , then a pattern has a left-leaning entry in the
-th position; if
, a pattern has a right-leaning entry; and in the remaining cases the entry is generic. In the left-leaning case,
must be
, and in the right-leaning case,
must be
. In the generic case,
can be either
or
. Thus,
It proves (2), and we are done. QED.
Tokuyama’s formula is related to the representation theory of -adic groups as follows. Let
, where
is a local non-archimedean field with the uniformizer
and the cardinality of the residue field
. By Casselman-Shalika formula, the value of the spherical Whittaker function on the diagonal element
is given by
which up to a constant equals the right side of Tokuyama’s formula when . In other words, Tokuyama’s formula can be interpreted as a combinatorial evaluation of the spherical Whittaker function. Now, the Casselman-Shalika formula holds for other groups beside
evaluating the spherical Whittaker function as the product of the deformed Weyl denominator and the corresponding character.
For example, for the symplectic group , the formula, up to a constant, takes the form
A natural question is whether the formula has a combinatorial evaluation in terms of GT-patterns for the symplectic (and other) groups. For the symplectic group, the answer is affirmative and is achieved by Dmitriy Ivanov using the solvable lattice models. No short proof like the one presented above is known.
For the orthogonal groups, the answer is still unknown. In other words, it is unknown how to represent
as a weighted sum over orthogonal GT patterns.
Note that the idea of the short proof above doesn’t work (at least I tried and failed!) for symplectic or orthogonal groups: both the branching rules and Pieri’s rules become not multiplicity-free within the type.