Ph.D HBNI 2024

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 xλsμ/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 followsthat Tw0 (xλ sμ/γ (XΦ)) is a non-negative integral linear combination of Schur poly- nomials. Then the flagged skew LR coefficient c ν λ, 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 c ν λ, 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 coeffi- cients 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 c kν kλ, kμ/kγ (Φ) > 0 for some k ≥ 1 then c ν λ, 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.