Demazure Crystal Structure for Flagged Skew Tableaux and Flagged Reverse Plane Partitions [HBNI Th248]

Abstract

This thesis is divided into two parts. The first part includes the Demazure crystal structure for flagged reverse plane partitions and flagged skew semi-standard tableaux. The second part addresses the saturation property of the flagged skew Littlewood-Richardson (LR) coefficients. Let λ be a partition with at most n (n ≥ 1) parts and Sn be the symmetric group. We denote by Tab(λ, n) the set of all semi-standard tableaux of shape λ with entries ≤ n. The Demazure crystal Bw (λ) indexed by λ and w ∈ Sn is a certain subset of Tab(λ, n). In this thesis, every connected component of the crystal graph of the set of flagged reverse plane partitions is shown to be a Demazure crystal (upto isomorphism). As an important corollary, it provides an explicit decomposition of the flagged dual stable Grothendieck polynomial gλ/µ (XΦ ) into a non-negative integral linear combination of key polynomials. The Demazure crystal structure for flagged reverse plane partitions extends the Demazure crystal structure for flagged skew semi-standard tableaux. The earlier result lifts the key-positivity result [20,Theorem 20] of the flagged skew Schur polynomials sλ/µ (XΦ ) from character level to crystals. Given a skew shape λ/µ and a flag Φ, Reiner and Shimozono [20, Theorem 20] have given an explicit decomposition of the flagged skew Schur polynomial sλ/µ (XΦ ) into a non-negative integral linear combination of key polynomials. Then xλ sµ/γ (XΦ ) is also a non-negative integral linear combination of key polynomials by a theorem of Joseph [9, §2.11]. Let w0 be the longest permutation in Sn . Then it follows that Tw0 (xλ sµ/γ (XΦ )) is a non-negative integral linear combination of Schur polynomials. Then the flagged skew LR coefficient cλ,ν µ/γ (Φ) is the multiplicity of the Schur polynomial sν (x1 , x2 , . . . , xn ) in the expansion of Tw0 (xλ sµ/γ (XΦ )). When Φ = (n, n, . . . , n), these coefficients reduce to Zelevinsky’s extension [24] of the LR coefficients cλ,ν µ/γ defined by the multiplicity of sν (x1 , x2 , . . . , xn ) in the expansion of sλ (x1 , x2 , . . . , xn ) sµ/γ (x1 , x2 , . . . , xn ). These will reduce to the usual LR coefficients when we further take γ = (0, 0, . . . , 0). Then, in second part of the thesis, we will show the saturation theorem for these flagged skew LR coefficients namely if kννckλ,kµ/kγ (Φ) > 0 for some k ≥ 1 then cλ, µ/γ (Φ) > 0. Thus these coefficients have the “saturation property”, first established by Knutson and Tao [12] for the classical LR coefficients. We give a tableau model to compute the flagged skew LR coefficients using the Demazure crystal structure for flagged skew semi-standard tableaux. We then produce a hive model for these coefficients to conclude the saturation property.